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<?latexml package="mathtools"?>
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<section inlist="toc" xml:id="S1">
<tags>
<tag>1</tag>
<tag role="refnum">1</tag>
<tag role="typerefnum">§1</tag>
</tags>
<title><tag close=" ">1</tag>Sums and Limits</title>
<paragraph inlist="toc" xml:id="S1.SS0.SSS0.Px1">
<title>mathclap & friends</title>
<para xml:id="S1.SS0.SSS0.Px1.p1">
<equation xml:id="S1.Ex1">
<Math mode="display" tex="X=\sum_{1\leq i\leq j\leq n}X_{ij}" text="X = (sum _ (1 <= i <= j <= n))@(X _ (i * j))" xml:id="S1.Ex1.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">X</XMTok>
<XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="mid1"/>
<XMTok mathstyle="display" meaning="sum" role="SUMOP" scriptpos="mid">∑</XMTok>
<XMApp>
<XMTok meaning="multirelation"/>
<XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">i</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">j</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">n</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" role="UNKNOWN">X</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">i</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">j</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMApp>
</XMath>
</Math>
</equation>
<equation xml:id="S1.Ex2">
<Math mode="display" tex="X=\sum_{\mathclap{1\leq i\leq j\leq n}}X_{ij}" text="X = (sum _ (1 <= i <= j <= n))@(X _ (i * j))" xml:id="S1.Ex2.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">X</XMTok>
<XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="mid1"/>
<XMTok mathstyle="display" meaning="sum" role="SUMOP" scriptpos="mid">∑</XMTok>
<XMApp width="0pt" xoffset="-14.9pt">
<XMTok meaning="multirelation"/>
<XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">i</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">j</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">n</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" role="UNKNOWN">X</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">i</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">j</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMApp>
</XMath>
</Math>
</equation>
<equation xml:id="S1.Ex3">
<Math mode="display" tex="X=\sum_{\mathrlap{1\leq i\leq j\leq n}}X_{ij}" text="X = (sum _ (1 <= i <= j <= n))@(X _ (i * j))" xml:id="S1.Ex3.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">X</XMTok>
<XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="mid1"/>
<XMTok mathstyle="display" meaning="sum" role="SUMOP" scriptpos="mid">∑</XMTok>
<XMApp width="0pt">
<XMTok meaning="multirelation"/>
<XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">i</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">j</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">n</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" role="UNKNOWN">X</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">i</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">j</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMApp>
</XMath>
</Math>
</equation>
<equation xml:id="S1.Ex4">
<Math mode="display" tex="X=\sum_{\mathllap{1\leq i\leq j\leq n}}X_{ij}" text="X = (sum _ (1 <= i <= j <= n))@(X _ (i * j))" xml:id="S1.Ex4.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">X</XMTok>
<XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="mid1"/>
<XMTok mathstyle="display" meaning="sum" role="SUMOP" scriptpos="mid">∑</XMTok>
<XMApp width="0pt" xoffset="-29.7pt">
<XMTok meaning="multirelation"/>
<XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">i</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">j</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">n</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" role="UNKNOWN">X</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">i</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">j</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMApp>
</XMath>
</Math>
</equation>
</para>
</paragraph>
<paragraph inlist="toc" xml:id="S1.SS0.SSS0.Px2">
<title>Cramped</title>
<para xml:id="S1.SS0.SSS0.Px2.p1">
<equation xml:id="S1.Ex5">
<Math mode="display" tex="\cramped{x^{2}}\leftrightarrow x^{2}\quad\cramped[\scriptstyle]{x^{2}}% \leftrightarrow{\scriptstyle x^{2}}" text="x ^ 2 leftrightarrow list@(x ^ 2, x ^ 2) leftrightarrow x ^ 2" xml:id="S1.Ex5.m1">
<XMath>
<XMApp>
<XMTok meaning="multirelation"/>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
<XMTok name="leftrightarrow" role="METARELOP">↔</XMTok>
<XMDual>
<XMApp>
<XMTok meaning="list"/>
<XMRef idref="S1.Ex5.m1.1"/>
<XMRef idref="S1.Ex5.m1.2"/>
</XMApp>
<XMWrap>
<XMApp xml:id="S1.Ex5.m1.1">
<XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMApp xml:id="S1.Ex5.m1.2">
<XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">x</XMTok>
<XMTok fontsize="50%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
</XMWrap>
</XMDual>
<XMTok fontsize="70%" name="leftrightarrow" role="METARELOP">↔</XMTok>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">x</XMTok>
<XMTok fontsize="50%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
</XMApp>
</XMath>
</Math>
</equation>
</para>
</paragraph>
<paragraph inlist="toc" xml:id="S1.SS0.SSS0.Px3">
<title>Smashoperator</title>
<para xml:id="S1.SS0.SSS0.Px3.p1">
<equation xml:id="S1.Ex6">
<Math mode="display" tex="V=\sum_{1\leq i\leq j\leq n}^{\infty}V_{ij}\quad X=\sum_{1\leq i\leq j\leq n}^% {3456}X_{ij}\quad Y=\sum\limits_{1\leq i\leq j\leq n}Y_{ij}\quad Z=\mathop{T}_% {1\leq i\leq j\leq n}Z_{ij}" text="formulae@(V = ((sum _ (1 <= i <= j <= n)) ^ infinity)@(V _ (i * j)), formulae@(X = ((sum _ (1 <= i <= j <= n)) ^ 3456)@(X _ (i * j)), formulae@(Y = (sum _ (1 <= i <= j <= n))@(Y _ (i * j)), Z = (T _ (1 <= i <= j <= n))@(Z _ (i * j)))))" xml:id="S1.Ex6.m1">
<XMath>
<XMDual>
<XMApp>
<XMTok meaning="formulae"/>
<XMRef idref="S1.Ex6.m1.1"/>
<XMRef idref="S1.Ex6.m1.2"/>
</XMApp>
<XMWrap>
<XMApp xml:id="S1.Ex6.m1.1">
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">V</XMTok>
<XMApp>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="mid1"/>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="mid1"/>
<XMTok mathstyle="display" meaning="sum" role="SUMOP" scriptpos="mid">∑</XMTok>
<XMApp>
<XMTok meaning="multirelation"/>
<XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">i</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">j</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">n</XMTok>
</XMApp>
</XMApp>
<XMTok fontsize="70%" meaning="infinity" name="infty" role="ID">∞</XMTok>
</XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" role="UNKNOWN">V</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">i</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">j</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMApp>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMDual xml:id="S1.Ex6.m1.2">
<XMApp>
<XMTok meaning="formulae"/>
<XMRef idref="S1.Ex6.m1.2.1"/>
<XMRef idref="S1.Ex6.m1.2.2"/>
</XMApp>
<XMWrap>
<XMApp xml:id="S1.Ex6.m1.2.1">
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">X</XMTok>
<XMApp>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="mid1"/>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="mid1"/>
<XMTok mathstyle="display" meaning="sum" role="SUMOP" scriptpos="mid">∑</XMTok>
<XMApp>
<XMTok meaning="multirelation"/>
<XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">i</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">j</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">n</XMTok>
</XMApp>
</XMApp>
<XMTok fontsize="70%" meaning="3456" role="NUMBER">3456</XMTok>
</XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" role="UNKNOWN">X</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">i</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">j</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMApp>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMDual xml:id="S1.Ex6.m1.2.2">
<XMApp>
<XMTok meaning="formulae"/>
<XMRef idref="S1.Ex6.m1.2.2.1"/>
<XMRef idref="S1.Ex6.m1.2.2.2"/>
</XMApp>
<XMWrap>
<XMApp xml:id="S1.Ex6.m1.2.2.1">
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">Y</XMTok>
<XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="mid1"/>
<XMTok mathstyle="display" meaning="sum" role="SUMOP" scriptpos="mid1">∑</XMTok>
<XMApp>
<XMTok meaning="multirelation"/>
<XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">i</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">j</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">n</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" role="UNKNOWN">Y</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">i</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">j</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMApp>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMApp xml:id="S1.Ex6.m1.2.2.2">
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">Z</XMTok>
<XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="mid1"/>
<XMTok font="italic" role="BIGOP" scriptpos="mid">T</XMTok>
<XMApp>
<XMTok meaning="multirelation"/>
<XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">i</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">j</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">n</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" role="UNKNOWN">Z</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">i</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">j</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMApp>
</XMWrap>
</XMDual>
</XMWrap>
</XMDual>
</XMWrap>
</XMDual>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S1.SS0.SSS0.Px3.p2">
<equation xml:id="S1.Ex7">
<Math mode="display" tex="V=\sum_{1\leq i\leq j\leq n}^{\infty}V_{ij}\quad X=\smashoperator[]{\sum_{1% \leq i\leq j\leq n}^{3456}}X_{ij}\quad Y=\smashoperator[r]{\sum_{1\leq i\leq j% \leq n}^{}}Y_{ij}\quad Z=\smashoperator[l]{\mathop{T}_{1\leq i\leq j\leq n}^{}% }Z_{ij}" text="formulae@(V = ((sum _ (1 <= i <= j <= n)) ^ infinity)@(V _ (i * j)), formulae@(X = ((sum ^ 3456) _ (1 <= i <= j <= n))@(X _ (i * j)), formulae@(Y = (sum _ (1 <= i <= j <= n))@(Y _ (i * j)), Z = (T _ (1 <= i <= j <= n))@(Z _ (i * j)))))" xml:id="S1.Ex7.m1">
<XMath>
<XMDual>
<XMApp>
<XMTok meaning="formulae"/>
<XMRef idref="S1.Ex7.m1.1"/>
<XMRef idref="S1.Ex7.m1.2"/>
</XMApp>
<XMWrap>
<XMApp xml:id="S1.Ex7.m1.1">
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">V</XMTok>
<XMApp>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="mid1"/>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="mid1"/>
<XMTok mathstyle="display" meaning="sum" role="SUMOP" scriptpos="mid">∑</XMTok>
<XMApp>
<XMTok meaning="multirelation"/>
<XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">i</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">j</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">n</XMTok>
</XMApp>
</XMApp>
<XMTok fontsize="70%" meaning="infinity" name="infty" role="ID">∞</XMTok>
</XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" role="UNKNOWN">V</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">i</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">j</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMApp>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMDual xml:id="S1.Ex7.m1.2">
<XMApp>
<XMTok meaning="formulae"/>
<XMRef idref="S1.Ex7.m1.2.1"/>
<XMRef idref="S1.Ex7.m1.2.2"/>
</XMApp>
<XMWrap>
<XMApp xml:id="S1.Ex7.m1.2.1">
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">X</XMTok>
<XMApp>
<XMApp role="SUMOP" scriptpos="mid">
<XMTok role="SUBSCRIPTOP" scriptpos="mid1"/>
<XMApp scriptpos="mid">
<XMTok role="SUPERSCRIPTOP" scriptpos="mid1"/>
<XMTok mathstyle="display" meaning="sum" role="SUMOP" scriptpos="mid">∑</XMTok>
<XMTok fontsize="70%" meaning="3456" role="NUMBER" width="0.0pt" xoffset="-7.0pt">3456</XMTok>
</XMApp>
<XMApp width="0.0pt" xoffset="-14.9pt">
<XMTok meaning="multirelation"/>
<XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">i</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">j</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">n</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" role="UNKNOWN">X</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">i</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">j</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMApp>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMDual xml:id="S1.Ex7.m1.2.2">
<XMApp>
<XMTok meaning="formulae"/>
<XMRef idref="S1.Ex7.m1.2.2.1"/>
<XMRef idref="S1.Ex7.m1.2.2.2"/>
</XMApp>
<XMWrap>
<XMApp xml:id="S1.Ex7.m1.2.2.1">
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">Y</XMTok>
<XMApp>
<XMApp lpadding="9.6pt" role="SUMOP" scriptpos="mid">
<XMTok role="SUBSCRIPTOP" scriptpos="mid1"/>
<XMTok mathstyle="display" meaning="sum" role="SUMOP" scriptpos="mid">∑</XMTok>
<XMApp width="0.0pt" xoffset="-14.9pt">
<XMTok meaning="multirelation"/>
<XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">i</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">j</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">n</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" role="UNKNOWN">Y</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">i</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">j</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMApp>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMApp xml:id="S1.Ex7.m1.2.2.2">
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">Z</XMTok>
<XMApp>
<XMApp role="SUMOP" rpadding="11.2pt" scriptpos="mid">
<XMTok role="SUBSCRIPTOP" scriptpos="mid1"/>
<XMTok font="italic" role="BIGOP" scriptpos="mid">T</XMTok>
<XMApp width="0.0pt" xoffset="-14.9pt">
<XMTok meaning="multirelation"/>
<XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">i</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">j</XMTok>
<XMTok fontsize="70%" meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">n</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" role="UNKNOWN">Z</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">i</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">j</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMApp>
</XMWrap>
</XMDual>
</XMWrap>
</XMDual>
</XMWrap>
</XMDual>
</XMath>
</Math>
</equation>
</para>
</paragraph>
<paragraph inlist="toc" xml:id="S1.SS0.SSS0.Px4">
<title>Adjustlimits</title>
<para xml:id="S1.SS0.SSS0.Px4.p1">
<equation xml:id="S1.Ex8">
<Math mode="display" tex="\text{a)}\lim_{n\to\infty}\max_{p\geq n}\quad\text{b)}\lim_{n\to\infty}\max_{p% ^{2}\geq n}\quad\text{c)}\lim_{n\to\infty}\sup_{p^{2}\geq nK}\quad\text{d)}% \limsup_{n\to\infty}\max_{p\geq n}" text="list@([a)] * (limit _ (n to infinity))@(maximum _ (p >= n)), [b)] * (limit _ (n to infinity))@(maximum _ (p ^ 2 >= n)), [c)] * (limit _ (n to infinity))@(supremum _ (p ^ 2 >= n * K)), [d)] * (limit-supremum _ (n to infinity))@(maximum _ (p >= n)))" xml:id="S1.Ex8.m1">
<XMath>
<XMDual>
<XMApp>
<XMTok meaning="list"/>
<XMRef idref="S1.Ex8.m1.1"/>
<XMRef idref="S1.Ex8.m1.2"/>
<XMRef idref="S1.Ex8.m1.3"/>
<XMRef idref="S1.Ex8.m1.4"/>
</XMApp>
<XMWrap>
<XMApp xml:id="S1.Ex8.m1.1">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMText>a)</XMText>
<XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="mid1"/>
<XMTok meaning="limit" role="LIMITOP" scriptpos="mid">lim</XMTok>
<XMApp>
<XMTok fontsize="70%" name="to" role="ARROW">→</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">n</XMTok>
<XMTok fontsize="70%" meaning="infinity" name="infty" role="ID">∞</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="mid1"/>
<XMTok meaning="maximum" role="OPFUNCTION" scriptpos="mid">max</XMTok>
<XMApp>
<XMTok fontsize="70%" meaning="greater-than-or-equals" name="geq" role="RELOP">≥</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">p</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">n</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMApp>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMApp xml:id="S1.Ex8.m1.2">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMText>b)</XMText>
<XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="mid1"/>
<XMTok meaning="limit" role="LIMITOP" scriptpos="mid">lim</XMTok>
<XMApp>
<XMTok fontsize="70%" name="to" role="ARROW">→</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">n</XMTok>
<XMTok fontsize="70%" meaning="infinity" name="infty" role="ID">∞</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="mid1"/>
<XMTok meaning="maximum" role="OPFUNCTION" scriptpos="mid">max</XMTok>
<XMApp>
<XMTok fontsize="70%" meaning="greater-than-or-equals" name="geq" role="RELOP">≥</XMTok>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">p</XMTok>
<XMTok fontsize="50%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">n</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMApp>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMApp xml:id="S1.Ex8.m1.3">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMText>c)</XMText>
<XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="mid1"/>
<XMTok meaning="limit" role="LIMITOP" scriptpos="mid">lim</XMTok>
<XMApp>
<XMTok fontsize="70%" name="to" role="ARROW">→</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">n</XMTok>
<XMTok fontsize="70%" meaning="infinity" name="infty" role="ID">∞</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="mid1"/>
<XMTok meaning="supremum" role="LIMITOP" scriptpos="mid">sup</XMTok>
<XMApp>
<XMTok fontsize="70%" meaning="greater-than-or-equals" name="geq" role="RELOP">≥</XMTok>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">p</XMTok>
<XMTok fontsize="50%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">n</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">K</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMApp>
</XMApp>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMApp xml:id="S1.Ex8.m1.4">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMText>d)</XMText>
<XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="mid1"/>
<XMTok meaning="limit-supremum" name="limsup" role="LIMITOP" scriptpos="mid">lim sup</XMTok>
<XMApp>
<XMTok fontsize="70%" name="to" role="ARROW">→</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">n</XMTok>
<XMTok fontsize="70%" meaning="infinity" name="infty" role="ID">∞</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="mid1"/>
<XMTok meaning="maximum" role="OPFUNCTION" scriptpos="mid">max</XMTok>
<XMApp>
<XMTok fontsize="70%" meaning="greater-than-or-equals" name="geq" role="RELOP">≥</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">p</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">n</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMApp>
</XMWrap>
</XMDual>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S1.SS0.SSS0.Px4.p2">
<equation xml:id="S1.Ex9">
<Math mode="display" tex="\text{a)}\adjustlimits{\lim}_{n\to\infty}{\max}_{p\geq n}\quad\text{b)}% \adjustlimits{\lim}_{n\to\infty}{\max}_{p^{2}\geq n}\quad\text{c)}% \adjustlimits{\lim}_{n\to\infty}{\sup}_{p^{2}\geq nK}\quad\text{d)}% \adjustlimits{\limsup}_{n\to\infty}{\max}_{p\geq n}" text="list@([a)] * limit _ (n to infinity) * maximum _ (p >= n), [b)] * limit _ (n to infinity) * maximum _ (p ^ 2 >= n), [c)] * limit _ (n to infinity) * supremum _ (p ^ 2 >= n * K), [d)] * limit-supremum _ (n to infinity) * maximum _ (p >= n))" xml:id="S1.Ex9.m1">
<XMath>
<XMDual>
<XMApp>
<XMTok meaning="list"/>
<XMRef idref="S1.Ex9.m1.1"/>
<XMRef idref="S1.Ex9.m1.2"/>
<XMRef idref="S1.Ex9.m1.3"/>
<XMRef idref="S1.Ex9.m1.4"/>
</XMApp>
<XMWrap>
<XMApp xml:id="S1.Ex9.m1.1">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMText>a)</XMText>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="mid"/>
<XMTok depth="0.0pt" meaning="limit" role="LIMITOP" scriptpos="mid">lim</XMTok>
<XMApp height="5.0pt">
<XMTok fontsize="70%" name="to" role="ARROW">→</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">n</XMTok>
<XMTok fontsize="70%" meaning="infinity" name="infty" role="ID">∞</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="mid"/>
<XMTok depth="0.0pt" meaning="maximum" role="OPFUNCTION" scriptpos="mid">max</XMTok>
<XMApp height="5.0pt">
<XMTok fontsize="70%" meaning="greater-than-or-equals" name="geq" role="RELOP">≥</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">p</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">n</XMTok>
</XMApp>
</XMApp>
</XMApp>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMApp xml:id="S1.Ex9.m1.2">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMText>b)</XMText>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="mid"/>
<XMTok depth="0.0pt" meaning="limit" role="LIMITOP" scriptpos="mid">lim</XMTok>
<XMApp height="5.0pt">
<XMTok fontsize="70%" name="to" role="ARROW">→</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">n</XMTok>
<XMTok fontsize="70%" meaning="infinity" name="infty" role="ID">∞</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="mid"/>
<XMTok depth="0.0pt" meaning="maximum" role="OPFUNCTION" scriptpos="mid">max</XMTok>
<XMApp height="5.0pt">
<XMTok fontsize="70%" meaning="greater-than-or-equals" name="geq" role="RELOP">≥</XMTok>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">p</XMTok>
<XMTok fontsize="50%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">n</XMTok>
</XMApp>
</XMApp>
</XMApp>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMApp xml:id="S1.Ex9.m1.3">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMText>c)</XMText>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="mid"/>
<XMTok depth="1.9pt" meaning="limit" role="LIMITOP" scriptpos="mid">lim</XMTok>
<XMApp height="5.0pt">
<XMTok fontsize="70%" name="to" role="ARROW">→</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">n</XMTok>
<XMTok fontsize="70%" meaning="infinity" name="infty" role="ID">∞</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="mid"/>
<XMTok depth="1.9pt" meaning="supremum" role="LIMITOP" scriptpos="mid">sup</XMTok>
<XMApp height="5.0pt">
<XMTok fontsize="70%" meaning="greater-than-or-equals" name="geq" role="RELOP">≥</XMTok>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">p</XMTok>
<XMTok fontsize="50%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">n</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">K</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMApp>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMApp xml:id="S1.Ex9.m1.4">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMText>d)</XMText>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="mid"/>
<XMTok depth="1.9pt" meaning="limit-supremum" name="limsup" role="LIMITOP" scriptpos="mid">lim sup</XMTok>
<XMApp height="5.0pt">
<XMTok fontsize="70%" name="to" role="ARROW">→</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">n</XMTok>
<XMTok fontsize="70%" meaning="infinity" name="infty" role="ID">∞</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="mid"/>
<XMTok depth="1.9pt" meaning="maximum" role="OPFUNCTION" scriptpos="mid">max</XMTok>
<XMApp height="5.0pt">
<XMTok fontsize="70%" meaning="greater-than-or-equals" name="geq" role="RELOP">≥</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">p</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">n</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMWrap>
</XMDual>
</XMath>
</Math>
</equation>
</para>
</paragraph>
</section>
<section inlist="toc" xml:id="S2">
<tags>
<tag>2</tag>
<tag role="refnum">2</tag>
<tag role="typerefnum">§2</tag>
</tags>
<title><tag close=" ">2</tag>Tags</title>
<para xml:id="S2.p1">
<quote>
<equation labels="LABEL:eq:example" xml:id="S2.Ex10">
<tags>
<tag>Q&A</tag>
<tag role="refnum">Q&A</tag>
</tags>
<Math mode="display" tex="a=b" text="a = b" xml:id="S2.Ex10.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
</XMApp>
</XMath>
</Math>
</equation>
<p><text font="italic">See <ref labelref="LABEL:eq:example"/> or is it better with <ref labelref="LABEL:eq:example"/>?
In the star form <verbatim font="typewriter upright">\ref*</verbatim> becomes <ref class="ltx_nolink" labelref="LABEL:eq:example"/> (<verbatim font="typewriter upright">\refeq*</verbatim> is not defined).</text></p>
</quote>
</para>
<para xml:id="S2.p2">
<equation labels="LABEL:eq:example2" xml:id="S2.Ex11">
<tags>
<tag>Q&A</tag>
<tag role="refnum">Q&A</tag>
</tags>
<Math mode="display" tex="a=b" text="a = b" xml:id="S2.Ex11.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
</XMApp>
</XMath>
</Math>
</equation>
<equation labels="LABEL:eq:example3" xml:id="S2.Ex12">
<tags>
<tag>[<text font="bold">Q&A</text>]</tag>
<tag role="refnum">Q&A</tag>
</tags>
<Math mode="display" tex="a=b" text="a = b" xml:id="S2.Ex12.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
</XMApp>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S2.p3">
<p>Normal tags.</p>
<equation labels="LABEL:eq:a0" xml:id="S2.E1">
<tags>
<tag>(1)</tag>
<tag role="refnum">1</tag>
</tags>
<Math mode="display" tex="a=a" text="a = a" xml:id="S2.E1.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
</XMApp>
</XMath>
</Math>
</equation>
<p>That was equation (<ref labelref="LABEL:eq:a0"/>).</p>
</para>
<para xml:id="S2.p4">
<p>OK tags.</p>
<equation labels="LABEL:eq:a1" xml:id="S2.E2">
<tags>
<tag>[<text font="bold">2</text>]</tag>
<tag role="refnum">2</tag>
</tags>
<Math mode="display" tex="a=a" text="a = a" xml:id="S2.E2.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
</XMApp>
</XMath>
</Math>
</equation>
<p>That was equation [<ref font="bold" labelref="LABEL:eq:a1"/>], but recall [<ref font="bold" labelref="LABEL:eq:a0"/>]</p>
</para>
<para xml:id="S2.p5">
<p>odd tag.</p>
<equation labels="LABEL:eq:a2" xml:id="S2.E3">
<tags>
<tag>{3}</tag>
<tag role="refnum">3</tag>
</tags>
<Math mode="display" tex="a=a" text="a = a" xml:id="S2.E3.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
</XMApp>
</XMath>
</Math>
</equation>
<p>That was equation {<ref labelref="LABEL:eq:a2"/>}, but recall {<ref labelref="LABEL:eq:a0"/>} and {<ref labelref="LABEL:eq:a1"/>}.</p>
</para>
<para xml:id="S2.p6">
<p>weird tag.</p>
<equation labels="LABEL:eq:b1" xml:id="S2.E4">
<tags>
<tag>((4))</tag>
<tag role="refnum">4</tag>
</tags>
<Math mode="display" tex="b=b" text="b = b" xml:id="S2.E4.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
</XMApp>
</XMath>
</Math>
</equation>
<p>That was equation ((<ref labelref="LABEL:eq:b1"/>)), but recall ((<ref labelref="LABEL:eq:a0"/>)), ((<ref labelref="LABEL:eq:a1"/>)) and ((<ref labelref="LABEL:eq:a2"/>)).</p>
</para>
<para xml:id="S2.p7">
<p>Normal tags again.</p>
<equation labels="LABEL:eq:c0" xml:id="S2.E5">
<tags>
<tag>(5)</tag>
<tag role="refnum">5</tag>
</tags>
<Math mode="display" tex="c=c" text="c = c" xml:id="S2.E5.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
</XMApp>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S2.p8">
<p>Non-textual</p>
<equation labels="LABEL:eq:nt1" xml:id="S2.Ex13">
<tags>
<tag>(n<sup><text font="italic">th</text></sup>)</tag>
<tag role="refnum">n<sup><text font="italic">th</text></sup></tag>
</tags>
<Math mode="display" tex="d=d" text="d = d" xml:id="S2.Ex13.m3">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
</XMApp>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S2.p9">
<p>That was equation (<ref labelref="LABEL:eq:c0"/>), but recall (<ref labelref="LABEL:eq:a0"/>), (<ref labelref="LABEL:eq:a1"/>), (<ref labelref="LABEL:eq:a2"/>),
(<ref labelref="LABEL:eq:b1"/>) and (<ref labelref="LABEL:eq:nt1"/>).</p>
</para>
<para xml:id="S2.p10">
<equationgroup class="ltx_eqn_gather" xml:id="S12.EGx1">
<equation labels="LABEL:eq:a" xml:id="S2.E6">
<tags>
<tag>(6)</tag>
<tag role="refnum">6</tag>
</tags>
<Math mode="display" tex="\displaystyle a=a" text="a = a" xml:id="S2.E6.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
</XMApp>
</XMath>
</Math>
</equation>
<equation labels="LABEL:eq:b" xml:id="S2.Ex14">
<tags>
<tag>(**)</tag>
<tag role="refnum">**</tag>
</tags>
<Math mode="display" tex="\displaystyle b=b" text="b = b" xml:id="S2.Ex14.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
</XMApp>
</XMath>
</Math>
</equation>
</equationgroup>
<p>This should refer to the equation containing <Math mode="inline" tex="a=a" text="a = a" xml:id="S2.p10.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
</XMApp>
</XMath>
</Math>: (<ref labelref="LABEL:eq:a"/>).
Then a switch of tag forms.</p>
<equationgroup class="ltx_eqn_align" xml:id="S12.EGx2">
<equation labels="LABEL:eq:c" xml:id="S2.E7">
<tags>
<tag>(7)</tag>
<tag role="refnum">7</tag>
</tags>
<MathFork>
<Math tex="\displaystyle c=c" text="c = c" xml:id="S2.E7.m3">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
</XMApp>
</XMath>
</Math>
<MathBranch>
<td align="right"><Math mode="inline" tex="\displaystyle c" text="c" xml:id="S2.E7.m1">
<XMath>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
</XMath>
</Math></td>
<td align="left"><Math mode="inline" tex="\displaystyle=c" text="absent = c" xml:id="S2.E7.m2">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok meaning="absent"/>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
</XMApp>
</XMath>
</Math></td>
</MathBranch>
</MathFork>
</equation>
<equation labels="LABEL:eq:d" xml:id="S2.E8">
<tags>
<tag>(8)</tag>
<tag role="refnum">8</tag>
</tags>
<MathFork>
<Math tex="\displaystyle d=d" text="d = d" xml:id="S2.E8.m3">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
</XMApp>
</XMath>
</Math>
<MathBranch>
<td align="right"><Math mode="inline" tex="\displaystyle d" text="d" xml:id="S2.E8.m1">
<XMath>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
</XMath>
</Math></td>
<td align="left"><Math mode="inline" tex="\displaystyle=d" text="absent = d" xml:id="S2.E8.m2">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok meaning="absent"/>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
</XMApp>
</XMath>
</Math></td>
</MathBranch>
</MathFork>
</equation>
</equationgroup>
<p>This should refer to the equation containing <Math mode="inline" tex="d=d" text="d = d" xml:id="S2.p10.m2">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
</XMApp>
</XMath>
</Math>: (<ref labelref="LABEL:eq:d"/>)
(but recall (<ref labelref="LABEL:eq:a"/>)).</p>
<equation xml:id="S2.E9">
<tags>
<tag>(9)</tag>
<tag role="refnum">9</tag>
</tags>
<Math mode="display" tex="e=e" text="e = e" xml:id="S2.E9.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
</XMApp>
</XMath>
</Math>
</equation>
<equation xml:id="S2.E10">
<tags>
<tag>(10)</tag>
<tag role="refnum">10</tag>
</tags>
<Math mode="display" tex="f=f" text="f = f" xml:id="S2.E10.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">f</XMTok>
<XMTok font="italic" role="UNKNOWN">f</XMTok>
</XMApp>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S2.p11">
<equation labels="LABEL:eq:much" xml:id="S2.Ex15">
<Math mode="display" tex="1+1=2" text="1 + 1 = 2" xml:id="S2.Ex15.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMTok meaning="1" role="NUMBER">1</XMTok>
<XMTok meaning="1" role="NUMBER">1</XMTok>
</XMApp>
<XMTok meaning="2" role="NUMBER">2</XMTok>
</XMApp>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S2.p12">
<equation labels="LABEL:eq:wow" xml:id="S2.Ex16">
<Math mode="display" tex="2+2=4" text="2 + 2 = 4" xml:id="S2.Ex16.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMTok meaning="2" role="NUMBER">2</XMTok>
<XMTok meaning="2" role="NUMBER">2</XMTok>
</XMApp>
<XMTok meaning="4" role="NUMBER">4</XMTok>
</XMApp>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S2.p13">
<p>Blabla (<ref labelref="LABEL:eq:wow"/>).</p>
</para>
</section>
<section inlist="toc" xml:id="S3">
<tags>
<tag>3</tag>
<tag role="refnum">3</tag>
<tag role="typerefnum">§3</tag>
</tags>
<title><tag close=" ">3</tag>Arrows</title>
<para xml:id="S3.p1">
<equation xml:id="S3.Ex17">
<Math mode="display" tex="A\xLeftarrow[under]{over}B\xRightarrow[under]{over}C" text="A (u * n * d * e * r)@((o * v * e * r)@(xLeftarrow)) B (u * n * d * e * r)@((o * v * e * r)@(xRightarrow)) C" xml:id="S3.Ex17.m1">
<XMath>
<XMApp>
<XMTok meaning="multirelation"/>
<XMTok font="italic" role="UNKNOWN">A</XMTok>
<XMApp role="ARROW">
<XMApp role="UNDERACCENT">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">u</XMTok>
<XMTok font="italic" role="UNKNOWN">n</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
<XMTok font="italic" role="UNKNOWN">r</XMTok>
</XMApp>
<XMApp>
<XMApp role="OVERACCENT">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
<XMTok font="italic" role="UNKNOWN">v</XMTok>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
<XMTok font="italic" role="UNKNOWN">r</XMTok>
</XMApp>
<XMTok name="xLeftarrow" role="ARROW" stretchy="true">⇐</XMTok>
</XMApp>
</XMApp>
<XMTok font="italic" role="UNKNOWN">B</XMTok>
<XMApp role="ARROW">
<XMApp role="UNDERACCENT">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">u</XMTok>
<XMTok font="italic" role="UNKNOWN">n</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
<XMTok font="italic" role="UNKNOWN">r</XMTok>
</XMApp>
<XMApp>
<XMApp role="OVERACCENT">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
<XMTok font="italic" role="UNKNOWN">v</XMTok>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
<XMTok font="italic" role="UNKNOWN">r</XMTok>
</XMApp>
<XMTok name="xRightarrow" role="ARROW" stretchy="true">⇒</XMTok>
</XMApp>
</XMApp>
<XMTok font="italic" role="UNKNOWN">C</XMTok>
</XMApp>
</XMath>
</Math>
</equation>
<equation xml:id="S3.Ex18">
<Math mode="display" tex="x\xleftrightarrow[under]{overlooooooong}y\xLeftrightarrow[underloooooooong]{% over}z" text="x (u * n * d * e * r)@((o * v * e * r * l * o * o * o * o * o * o * o * n * g)@(xleftrightarrow)) y (u * n * d * e * r * l * o * o * o * o * o * o * o * o * n * g)@((o * v * e * r)@(xLeftrightarrow)) z" xml:id="S3.Ex18.m1">
<XMath>
<XMApp>
<XMApp role="METARELOP">
<XMApp role="UNDERACCENT">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">u</XMTok>
<XMTok font="italic" role="UNKNOWN">n</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
<XMTok font="italic" role="UNKNOWN">r</XMTok>
</XMApp>
<XMApp>
<XMApp role="OVERACCENT">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
<XMTok font="italic" role="UNKNOWN">v</XMTok>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
<XMTok font="italic" role="UNKNOWN">r</XMTok>
<XMTok font="italic" role="UNKNOWN">l</XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
<XMTok font="italic" role="UNKNOWN">n</XMTok>
<XMTok font="italic" role="UNKNOWN">g</XMTok>
</XMApp>
<XMTok name="xleftrightarrow" role="METARELOP" stretchy="true">↔</XMTok>
</XMApp>
</XMApp>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
<XMApp>
<XMApp role="ARROW">
<XMApp role="UNDERACCENT">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">u</XMTok>
<XMTok font="italic" role="UNKNOWN">n</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
<XMTok font="italic" role="UNKNOWN">r</XMTok>
<XMTok font="italic" role="UNKNOWN">l</XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
<XMTok font="italic" role="UNKNOWN">n</XMTok>
<XMTok font="italic" role="UNKNOWN">g</XMTok>
</XMApp>
<XMApp>
<XMApp role="OVERACCENT">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
<XMTok font="italic" role="UNKNOWN">v</XMTok>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
<XMTok font="italic" role="UNKNOWN">r</XMTok>
</XMApp>
<XMTok name="xLeftrightarrow" role="ARROW" stretchy="true">⇔</XMTok>
</XMApp>
</XMApp>
<XMTok font="italic" role="UNKNOWN">y</XMTok>
<XMTok font="italic" role="UNKNOWN">z</XMTok>
</XMApp>
</XMApp>
</XMath>
</Math>
</equation>
<equation xml:id="S3.Ex19">
<Math mode="display" tex="x\xhookleftarrow[bar]{foo}y\xhookrightarrow[bluuuuuuuuub]{baz}t\xmapsto{% heeereee}k" text="x (b * a * r)@((f * o * o)@(xhookleftarrow)) y (b * l * u * u * u * u * u * u * u * u * u * b)@((b * a * z)@(xhookrightarrow)) t (h * e * e * e * r * e * e * e)@(xmapsto) k" xml:id="S3.Ex19.m1">
<XMath>
<XMApp>
<XMTok meaning="multirelation"/>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
<XMApp role="ARROW">
<XMApp role="UNDERACCENT">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">r</XMTok>
</XMApp>
<XMApp>
<XMApp role="OVERACCENT">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">f</XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
</XMApp>
<XMTok name="xhookleftarrow" role="ARROW" stretchy="true">↩</XMTok>
</XMApp>
</XMApp>
<XMTok font="italic" role="UNKNOWN">y</XMTok>
<XMApp role="ARROW">
<XMApp role="UNDERACCENT">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
<XMTok font="italic" role="UNKNOWN">l</XMTok>
<XMTok font="italic" role="UNKNOWN">u</XMTok>
<XMTok font="italic" role="UNKNOWN">u</XMTok>
<XMTok font="italic" role="UNKNOWN">u</XMTok>
<XMTok font="italic" role="UNKNOWN">u</XMTok>
<XMTok font="italic" role="UNKNOWN">u</XMTok>
<XMTok font="italic" role="UNKNOWN">u</XMTok>
<XMTok font="italic" role="UNKNOWN">u</XMTok>
<XMTok font="italic" role="UNKNOWN">u</XMTok>
<XMTok font="italic" role="UNKNOWN">u</XMTok>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
</XMApp>
<XMApp>
<XMApp role="OVERACCENT">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">z</XMTok>
</XMApp>
<XMTok name="xhookrightarrow" role="ARROW" stretchy="true">↪</XMTok>
</XMApp>
</XMApp>
<XMTok font="italic" role="UNKNOWN">t</XMTok>
<XMApp role="ARROW">
<XMApp role="OVERACCENT">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">h</XMTok>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
<XMTok font="italic" role="UNKNOWN">r</XMTok>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
</XMApp>
<XMTok name="xmapsto" role="ARROW">↦</XMTok>
</XMApp>
<XMTok font="italic" role="UNKNOWN">k</XMTok>
</XMApp>
</XMath>
</Math>
</equation>
<equation xml:id="S3.Ex20">
<Math class="ltx_math_unparsed" mode="display" tex="k\xleftharpoonup[.]{}l\xleftharpoondown{..}m\xrightharpoondown[...]{}n% \xrightharpoonup{....}o" text="k .@(absent@(xleftharpoonup)) l .@.@(xlefttharpoondown) m ldots@(absent@(xrightharpoondown)) n ldots@.@(xrightharpoonup) o" xml:id="S3.Ex20.m1">
<XMath>
<XMApp>
<XMTok meaning="multirelation"/>
<XMTok font="italic" role="UNKNOWN">k</XMTok>
<XMApp role="ARROW">
<XMTok role="UNDERACCENT">.</XMTok>
<XMApp>
<XMTok meaning="absent" role="OVERACCENT"/>
<XMTok name="xleftharpoonup" role="ARROW" stretchy="true">↼</XMTok>
</XMApp>
</XMApp>
<XMTok font="italic" role="UNKNOWN">l</XMTok>
<XMApp role="ARROW">
<XMWrap role="OVERACCENT">
<XMTok role="PERIOD">.</XMTok>
<XMTok role="PERIOD">.</XMTok>
</XMWrap>
<XMTok name="xlefttharpoondown" role="ARROW">↽</XMTok>
</XMApp>
<XMTok font="italic" role="UNKNOWN">m</XMTok>
<XMApp role="ARROW">
<XMTok name="ldots" role="UNDERACCENT">…</XMTok>
<XMApp>
<XMTok meaning="absent" role="OVERACCENT"/>
<XMTok name="xrightharpoondown" role="ARROW" stretchy="true">⇁</XMTok>
</XMApp>
</XMApp>
<XMTok font="italic" role="UNKNOWN">n</XMTok>
<XMApp role="ARROW">
<XMWrap role="OVERACCENT">
<XMTok name="ldots" role="ID">…</XMTok>
<XMTok role="PERIOD">.</XMTok>
</XMWrap>
<XMTok name="xrightharpoonup" role="ARROW">⇀</XMTok>
</XMApp>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
</XMApp>
</XMath>
</Math>
</equation>
<equation xml:id="S3.Ex21">
<Math mode="display" tex="x\xrightleftharpoons{bluuuuub}y\xleftrightharpoons[blaaaaaab]{}z" text="x (b * l * u * u * u * u * u * b)@(xrightleftharpoons) y (b * l * a * a * a * a * a * a * b)@(absent@(xleftrightharpoons)) z" xml:id="S3.Ex21.m1">
<XMath>
<XMApp>
<XMTok meaning="multirelation"/>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
<XMApp role="METARELOP">
<XMApp role="OVERACCENT">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
<XMTok font="italic" role="UNKNOWN">l</XMTok>
<XMTok font="italic" role="UNKNOWN">u</XMTok>
<XMTok font="italic" role="UNKNOWN">u</XMTok>
<XMTok font="italic" role="UNKNOWN">u</XMTok>
<XMTok font="italic" role="UNKNOWN">u</XMTok>
<XMTok font="italic" role="UNKNOWN">u</XMTok>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
</XMApp>
<XMTok name="xrightleftharpoons" role="METARELOP">⇌</XMTok>
</XMApp>
<XMTok font="italic" role="UNKNOWN">y</XMTok>
<XMApp role="METARELOP">
<XMApp role="UNDERACCENT">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
<XMTok font="italic" role="UNKNOWN">l</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
</XMApp>
<XMApp>
<XMTok meaning="absent" role="OVERACCENT"/>
<XMTok name="xleftrightharpoons" role="METARELOP" stretchy="true">⇋</XMTok>
</XMApp>
</XMApp>
<XMTok font="italic" role="UNKNOWN">z</XMTok>
</XMApp>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S3.p2">
<equation xml:id="S3.Ex22">
<Math mode="display" tex="z=\overbrace{\underbracket{x}_{\text{real}}+i\underbracket{y}_{\text{imaginary% }}}^{\text{complex number}}\quad\underbrace{1+1}_{=2}" text="z = list@((overbrace@((underbracket@(x)) _ [real] + i * (underbracket@(y)) _ [imaginary])) ^ [complex number], (underbrace@(1 + 1)) _ (absent = 2))" xml:id="S3.Ex22.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">z</XMTok>
<XMDual>
<XMApp>
<XMTok meaning="list"/>
<XMRef idref="S3.Ex22.m1.1"/>
<XMRef idref="S3.Ex22.m1.2"/>
</XMApp>
<XMWrap>
<XMApp xml:id="S3.Ex22.m1.1">
<XMTok role="SUPERSCRIPTOP" scriptpos="mid1"/>
<XMApp scriptpos="mid">
<XMTok name="overbrace" role="OVERACCENT">⏞</XMTok>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="mid2"/>
<XMApp scriptpos="mid">
<XMTok name="underbracket" role="UNDERACCENT">﹈</XMTok>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
</XMApp>
<XMText><text fontsize="70%">real</text></XMText>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">i</XMTok>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="mid2"/>
<XMApp scriptpos="mid">
<XMTok name="underbracket" role="UNDERACCENT">﹈</XMTok>
<XMTok font="italic" role="UNKNOWN">y</XMTok>
</XMApp>
<XMText><text fontsize="70%">imaginary</text></XMText>
</XMApp>
</XMApp>
</XMApp>
</XMApp>
<XMText><text fontsize="70%">complex number</text></XMText>
</XMApp>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMApp xml:id="S3.Ex22.m1.2">
<XMTok role="SUBSCRIPTOP" scriptpos="mid1"/>
<XMApp scriptpos="mid">
<XMTok name="underbrace" role="UNDERACCENT">⏟</XMTok>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMTok meaning="1" role="NUMBER">1</XMTok>
<XMTok meaning="1" role="NUMBER">1</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok fontsize="70%" meaning="equals" role="RELOP">=</XMTok>
<XMTok meaning="absent"/>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
</XMApp>
</XMWrap>
</XMDual>
</XMApp>
</XMath>
</Math>
</equation>
</para>
</section>
<section inlist="toc" xml:id="S4">
<tags>
<tag>4</tag>
<tag role="refnum">4</tag>
<tag role="typerefnum">§4</tag>
</tags>
<title><tag close=" ">4</tag>Matrices</title>
<para xml:id="S4.p1">
<equation xml:id="S4.Ex23">
<Math mode="display" tex="\begin{matrix}c&cocococococo\\ c&c\end{matrix}" text="matrix@(Array[[c, c * o * c * o * c * o * c * o * c * o * c * o], [c, c]])" xml:id="S4.Ex23.m1">
<XMath>
<XMDual>
<XMApp>
<XMTok meaning="matrix"/>
<XMRef idref="S4.Ex23.m1.1"/>
</XMApp>
<XMArray xml:id="S4.Ex23.m1.1">
<XMRow>
<XMCell align="center">
<XMTok font="italic" role="UNKNOWN">c</XMTok>
</XMCell>
<XMCell align="center">
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
</XMApp>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="center">
<XMTok font="italic" role="UNKNOWN">c</XMTok>
</XMCell>
<XMCell align="center">
<XMTok font="italic" role="UNKNOWN">c</XMTok>
</XMCell>
</XMRow>
</XMArray>
</XMDual>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S4.p2">
<equation xml:id="S4.Ex24">
<Math mode="display" tex="\begin{matrix}[l]lalalalalala&l\\ l&l\end{matrix}" text="matrix@(Array[[l * a * l * a * l * a * l * a * l * a * l * a, l], [l, l]])" xml:id="S4.Ex24.m1">
<XMath>
<XMDual>
<XMApp>
<XMTok meaning="matrix"/>
<XMRef idref="S4.Ex24.m1.1"/>
</XMApp>
<XMArray xml:id="S4.Ex24.m1.1">
<XMRow>
<XMCell align="left">
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">l</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">l</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">l</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">l</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">l</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">l</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
</XMApp>
</XMCell>
<XMCell align="left">
<XMTok font="italic" role="UNKNOWN">l</XMTok>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="left">
<XMTok font="italic" role="UNKNOWN">l</XMTok>
</XMCell>
<XMCell align="left">
<XMTok font="italic" role="UNKNOWN">l</XMTok>
</XMCell>
</XMRow>
</XMArray>
</XMDual>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S4.p3">
<equation xml:id="S4.Ex25">
<Math mode="display" tex="\begin{matrix}[r]rererererere&r\\ r&r\end{matrix}" text="matrix@(Array[[r * e * r * e * r * e * r * e * r * e * r * e, r], [r, r]])" xml:id="S4.Ex25.m1">
<XMath>
<XMDual>
<XMApp>
<XMTok meaning="matrix"/>
<XMRef idref="S4.Ex25.m1.1"/>
</XMApp>
<XMArray xml:id="S4.Ex25.m1.1">
<XMRow>
<XMCell align="right">
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">r</XMTok>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
<XMTok font="italic" role="UNKNOWN">r</XMTok>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
<XMTok font="italic" role="UNKNOWN">r</XMTok>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
<XMTok font="italic" role="UNKNOWN">r</XMTok>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
<XMTok font="italic" role="UNKNOWN">r</XMTok>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
<XMTok font="italic" role="UNKNOWN">r</XMTok>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
</XMApp>
</XMCell>
<XMCell align="right">
<XMTok font="italic" role="UNKNOWN">r</XMTok>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="right">
<XMTok font="italic" role="UNKNOWN">r</XMTok>
</XMCell>
<XMCell align="right">
<XMTok font="italic" role="UNKNOWN">r</XMTok>
</XMCell>
</XMRow>
</XMArray>
</XMDual>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S4.p4">
<equation xml:id="S4.Ex26">
<Math mode="display" tex="\begin{pmatrix}[l]ppppppp&foo\\ l&ppppppppppppppp\end{pmatrix}" text="matrix@(Array[[p * p * p * p * p * p * p, f * o * o], [l, p * p * p * p * p * p * p * p * p * p * p * p * p * p * p]])" xml:id="S4.Ex26.m1">
<XMath>
<XMDual>
<XMApp>
<XMTok meaning="matrix"/>
<XMRef idref="S4.Ex26.m1.1"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="true">(</XMTok>
<XMArray xml:id="S4.Ex26.m1.1">
<XMRow>
<XMCell align="left">
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">p</XMTok>
<XMTok font="italic" role="UNKNOWN">p</XMTok>
<XMTok font="italic" role="UNKNOWN">p</XMTok>
<XMTok font="italic" role="UNKNOWN">p</XMTok>
<XMTok font="italic" role="UNKNOWN">p</XMTok>
<XMTok font="italic" role="UNKNOWN">p</XMTok>
<XMTok font="italic" role="UNKNOWN">p</XMTok>
</XMApp>
</XMCell>
<XMCell align="left">
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">f</XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
</XMApp>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="left">
<XMTok font="italic" role="UNKNOWN">l</XMTok>
</XMCell>
<XMCell align="left">
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">p</XMTok>
<XMTok font="italic" role="UNKNOWN">p</XMTok>
<XMTok font="italic" role="UNKNOWN">p</XMTok>
<XMTok font="italic" role="UNKNOWN">p</XMTok>
<XMTok font="italic" role="UNKNOWN">p</XMTok>
<XMTok font="italic" role="UNKNOWN">p</XMTok>
<XMTok font="italic" role="UNKNOWN">p</XMTok>
<XMTok font="italic" role="UNKNOWN">p</XMTok>
<XMTok font="italic" role="UNKNOWN">p</XMTok>
<XMTok font="italic" role="UNKNOWN">p</XMTok>
<XMTok font="italic" role="UNKNOWN">p</XMTok>
<XMTok font="italic" role="UNKNOWN">p</XMTok>
<XMTok font="italic" role="UNKNOWN">p</XMTok>
<XMTok font="italic" role="UNKNOWN">p</XMTok>
<XMTok font="italic" role="UNKNOWN">p</XMTok>
</XMApp>
</XMCell>
</XMRow>
</XMArray>
<XMTok role="CLOSE" stretchy="true">)</XMTok>
</XMWrap>
</XMDual>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S4.p5">
<equation xml:id="S4.Ex27">
<Math mode="display" tex="\begin{bmatrix}b&b\\ b&b\end{bmatrix}" text="matrix@(Array[[b, b], [b, b]])" xml:id="S4.Ex27.m1">
<XMath>
<XMDual>
<XMApp>
<XMTok meaning="matrix"/>
<XMRef idref="S4.Ex27.m1.1"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="true">[</XMTok>
<XMArray xml:id="S4.Ex27.m1.1">
<XMRow>
<XMCell align="center">
<XMTok font="italic" role="UNKNOWN">b</XMTok>
</XMCell>
<XMCell align="center">
<XMTok font="italic" role="UNKNOWN">b</XMTok>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="center">
<XMTok font="italic" role="UNKNOWN">b</XMTok>
</XMCell>
<XMCell align="center">
<XMTok font="italic" role="UNKNOWN">b</XMTok>
</XMCell>
</XMRow>
</XMArray>
<XMTok role="CLOSE" stretchy="true">]</XMTok>
</XMWrap>
</XMDual>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S4.p6">
<equation xml:id="S4.Ex28">
<Math mode="display" tex="\begin{Bmatrix}[r]B&B\\ B&BBBBBBBrBBBBBB\end{Bmatrix}" text="matrix@(Array[[B, B], [B, B * B * B * B * B * B * B * r * B * B * B * B * B * B]])" xml:id="S4.Ex28.m1">
<XMath>
<XMDual>
<XMApp>
<XMTok meaning="matrix"/>
<XMRef idref="S4.Ex28.m1.1"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="true">{</XMTok>
<XMArray xml:id="S4.Ex28.m1.1">
<XMRow>
<XMCell align="right">
<XMTok font="italic" role="UNKNOWN">B</XMTok>
</XMCell>
<XMCell align="right">
<XMTok font="italic" role="UNKNOWN">B</XMTok>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="right">
<XMTok font="italic" role="UNKNOWN">B</XMTok>
</XMCell>
<XMCell align="right">
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">B</XMTok>
<XMTok font="italic" role="UNKNOWN">B</XMTok>
<XMTok font="italic" role="UNKNOWN">B</XMTok>
<XMTok font="italic" role="UNKNOWN">B</XMTok>
<XMTok font="italic" role="UNKNOWN">B</XMTok>
<XMTok font="italic" role="UNKNOWN">B</XMTok>
<XMTok font="italic" role="UNKNOWN">B</XMTok>
<XMTok font="italic" role="UNKNOWN">r</XMTok>
<XMTok font="italic" role="UNKNOWN">B</XMTok>
<XMTok font="italic" role="UNKNOWN">B</XMTok>
<XMTok font="italic" role="UNKNOWN">B</XMTok>
<XMTok font="italic" role="UNKNOWN">B</XMTok>
<XMTok font="italic" role="UNKNOWN">B</XMTok>
<XMTok font="italic" role="UNKNOWN">B</XMTok>
</XMApp>
</XMCell>
</XMRow>
</XMArray>
<XMTok role="CLOSE" stretchy="true">}</XMTok>
</XMWrap>
</XMDual>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S4.p7">
<equation xml:id="S4.Ex29">
<Math mode="display" tex="\begin{vmatrix}v&v\\ v&v\end{vmatrix}" text="determinant@(matrix@(Array[[v, v], [v, v]]))" xml:id="S4.Ex29.m1">
<XMath>
<XMDual>
<XMApp>
<XMTok meaning="determinant"/>
<XMApp>
<XMTok meaning="matrix"/>
<XMRef idref="S4.Ex29.m1.1"/>
</XMApp>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="true">|</XMTok>
<XMArray xml:id="S4.Ex29.m1.1">
<XMRow>
<XMCell align="center">
<XMTok font="italic" role="UNKNOWN">v</XMTok>
</XMCell>
<XMCell align="center">
<XMTok font="italic" role="UNKNOWN">v</XMTok>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="center">
<XMTok font="italic" role="UNKNOWN">v</XMTok>
</XMCell>
<XMCell align="center">
<XMTok font="italic" role="UNKNOWN">v</XMTok>
</XMCell>
</XMRow>
</XMArray>
<XMTok role="CLOSE" stretchy="true">|</XMTok>
</XMWrap>
</XMDual>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S4.p8">
<equation xml:id="S4.Ex30">
<Math mode="display" tex="\begin{Vmatrix}[c]V&V\\ VVVVVVcVVVVVV&bar\end{Vmatrix}" text="norm@(matrix@(Array[[V, V], [V * V * V * V * V * V * c * V * V * V * V * V * V, b * a * r]]))" xml:id="S4.Ex30.m1">
<XMath>
<XMDual>
<XMApp>
<XMTok meaning="norm"/>
<XMApp>
<XMTok meaning="matrix"/>
<XMRef idref="S4.Ex30.m1.1"/>
</XMApp>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="true">‖</XMTok>
<XMArray xml:id="S4.Ex30.m1.1">
<XMRow>
<XMCell align="center">
<XMTok font="italic" role="UNKNOWN">V</XMTok>
</XMCell>
<XMCell align="center">
<XMTok font="italic" role="UNKNOWN">V</XMTok>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="center">
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">V</XMTok>
<XMTok font="italic" role="UNKNOWN">V</XMTok>
<XMTok font="italic" role="UNKNOWN">V</XMTok>
<XMTok font="italic" role="UNKNOWN">V</XMTok>
<XMTok font="italic" role="UNKNOWN">V</XMTok>
<XMTok font="italic" role="UNKNOWN">V</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMTok font="italic" role="UNKNOWN">V</XMTok>
<XMTok font="italic" role="UNKNOWN">V</XMTok>
<XMTok font="italic" role="UNKNOWN">V</XMTok>
<XMTok font="italic" role="UNKNOWN">V</XMTok>
<XMTok font="italic" role="UNKNOWN">V</XMTok>
<XMTok font="italic" role="UNKNOWN">V</XMTok>
</XMApp>
</XMCell>
<XMCell align="center">
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">r</XMTok>
</XMApp>
</XMCell>
</XMRow>
</XMArray>
<XMTok role="CLOSE" stretchy="true">‖</XMTok>
</XMWrap>
</XMDual>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S4.p9">
<equation xml:id="S4.Ex31">
<Math mode="display" tex="\begin{vmatrix}[l]a&blblblbllbblblblblblblbl\\ c&d\end{vmatrix}" text="determinant@(matrix@(Array[[a, b * l * b * l * b * l * b * l * l * b * b * l * b * l * b * l * b * l * b * l * b * l * b * l], [c, d]]))" xml:id="S4.Ex31.m1">
<XMath>
<XMDual>
<XMApp>
<XMTok meaning="determinant"/>
<XMApp>
<XMTok meaning="matrix"/>
<XMRef idref="S4.Ex31.m1.1"/>
</XMApp>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="true">|</XMTok>
<XMArray xml:id="S4.Ex31.m1.1">
<XMRow>
<XMCell align="left">
<XMTok font="italic" fontsize="70%" role="UNKNOWN">a</XMTok>
</XMCell>
<XMCell align="left">
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">b</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">l</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">b</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">l</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">b</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">l</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">b</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">l</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">l</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">b</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">b</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">l</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">b</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">l</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">b</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">l</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">b</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">l</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">b</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">l</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">b</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">l</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">b</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">l</XMTok>
</XMApp>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="left">
<XMTok font="italic" fontsize="70%" role="UNKNOWN">c</XMTok>
</XMCell>
<XMCell align="left">
<XMTok font="italic" fontsize="70%" role="UNKNOWN">d</XMTok>
</XMCell>
</XMRow>
</XMArray>
<XMTok role="CLOSE" stretchy="true">|</XMTok>
</XMWrap>
</XMDual>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S4.p10">
<equation xml:id="S4.Ex32">
<Math mode="display" tex="\begin{bmatrix}a&-b\\ -c&d\end{bmatrix}\begin{bmatrix}[r]a&-b\\ -c&d\end{bmatrix}" text="matrix@(Array[[a, - b], [- c, d]]) * matrix@(Array[[a, - b], [- c, d]])" xml:id="S4.Ex32.m1">
<XMath>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMDual>
<XMApp>
<XMTok meaning="matrix"/>
<XMRef idref="S4.Ex32.m1.1"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="true">[</XMTok>
<XMArray xml:id="S4.Ex32.m1.1">
<XMRow>
<XMCell align="center">
<XMTok font="italic" fontsize="70%" role="UNKNOWN">a</XMTok>
</XMCell>
<XMCell align="center">
<XMApp>
<XMTok fontsize="70%" meaning="minus" role="ADDOP">-</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">b</XMTok>
</XMApp>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="center">
<XMApp>
<XMTok fontsize="70%" meaning="minus" role="ADDOP">-</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">c</XMTok>
</XMApp>
</XMCell>
<XMCell align="center">
<XMTok font="italic" fontsize="70%" role="UNKNOWN">d</XMTok>
</XMCell>
</XMRow>
</XMArray>
<XMTok role="CLOSE" stretchy="true">]</XMTok>
</XMWrap>
</XMDual>
<XMDual>
<XMApp>
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</section>
<section inlist="toc" xml:id="S5">
<tags>
<tag>5</tag>
<tag role="refnum">5</tag>
<tag role="typerefnum">§5</tag>
</tags>
<title><tag close=" ">5</tag>Cases</title>
<para xml:id="S5.p1">
<equation xml:id="S5.Ex38">
<Math mode="display" tex="\begin{cases}E=mc^{2}&\text{Nothing to see here}\\ \int x-3\,dx&\text{Integral is text style}\end{cases}" text="cases@(E = m * c ^ 2, [Nothing to see here], integral@(x) - 3 * d * x, [Integral is text style])" xml:id="S5.Ex38.m1">
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<XMRef idref="S5.Ex38.m1.4"/>
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<XMCell align="left">
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<XMRow>
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<XMTok meaning="minus" role="ADDOP">-</XMTok>
<XMApp>
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</XMApp>
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<XMCell align="left">
<XMText xml:id="S5.Ex38.m1.4">Integral is text style</XMText>
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</Math>
</equation>
</para>
<para xml:id="S5.p2">
<equation xml:id="S5.Ex39">
<Math mode="display" tex="\begin{dcases}E=mc^{2}&c\approx 3.00\times 10^{8}\,\mathrm{m}/\mathrm{s}\\ \int x-3\,dx&\text{Integral is display style}\end{dcases}" text="cases@(E = m * c ^ 2, c approximately-equals (3.00 * 10 ^ 8 * m) / s, integral@(x) - 3 * d * x, [Integral is display style])" xml:id="S5.Ex39.m1">
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</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="true">{</XMTok>
<XMArray>
<XMRow>
<XMCell align="left">
<XMApp xml:id="S5.Ex39.m1.1">
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">E</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">m</XMTok>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMCell>
<XMCell align="left">
<XMApp xml:id="S5.Ex39.m1.2">
<XMTok meaning="approximately-equals" name="approx" role="RELOP">≈</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMApp>
<XMTok meaning="divide" role="MULOP">/</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP">×</XMTok>
<XMTok meaning="3.00" role="NUMBER">3.00</XMTok>
<XMApp rpadding="1.7pt">
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMTok meaning="10" role="NUMBER">10</XMTok>
<XMTok fontsize="70%" meaning="8" role="NUMBER">8</XMTok>
</XMApp>
</XMApp>
<XMTok role="UNKNOWN">m</XMTok>
</XMApp>
<XMTok role="UNKNOWN">s</XMTok>
</XMApp>
</XMApp>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="left">
<XMApp xml:id="S5.Ex39.m1.3">
<XMTok meaning="minus" role="ADDOP">-</XMTok>
<XMApp>
<XMTok mathstyle="display" meaning="integral" name="int" role="INTOP">∫</XMTok>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok meaning="3" role="NUMBER" rpadding="1.7pt">3</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
</XMApp>
</XMApp>
</XMCell>
<XMCell align="left">
<XMText xml:id="S5.Ex39.m1.4">Integral is display style</XMText>
</XMCell>
</XMRow>
</XMArray>
</XMWrap>
</XMDual>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S5.p3">
<equation xml:id="S5.Ex40">
<Math mode="display" tex="a=\begin{dcases*}E=mc^{2}&Nothing to see here (text in math)\\ \int x-3\,dx&Integral is display style (text in math)\end{dcases*}" text="a = cases@(E = m * c ^ 2, [Nothing to see here (text in math)], integral@(x) - 3 * d * x, [Integral is display style (text in math)])" xml:id="S5.Ex40.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMDual>
<XMApp>
<XMTok meaning="cases"/>
<XMRef idref="S5.Ex40.m1.1"/>
<XMRef idref="S5.Ex40.m1.2"/>
<XMRef idref="S5.Ex40.m1.3"/>
<XMRef idref="S5.Ex40.m1.4"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="true">{</XMTok>
<XMArray>
<XMRow>
<XMCell align="left">
<XMApp xml:id="S5.Ex40.m1.1">
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">E</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">m</XMTok>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMCell>
<XMCell align="left">
<XMText xml:id="S5.Ex40.m1.2">Nothing to see here (text in math)</XMText>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="left">
<XMApp xml:id="S5.Ex40.m1.3">
<XMTok meaning="minus" role="ADDOP">-</XMTok>
<XMApp>
<XMTok mathstyle="display" meaning="integral" name="int" role="INTOP">∫</XMTok>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok meaning="3" role="NUMBER" rpadding="1.7pt">3</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
</XMApp>
</XMApp>
</XMCell>
<XMCell align="left">
<XMText xml:id="S5.Ex40.m1.4">Integral is display style (text in math)</XMText>
</XMCell>
</XMRow>
</XMArray>
</XMWrap>
</XMDual>
</XMApp>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S5.p4">
<equation xml:id="S5.Ex41">
<Math mode="display" tex="\begin{rcases}E=mc^{2}&5^{6}\quad\text{and so on}\\ \int x-3\,dx&\int x\,dx\end{rcases}=b" text="cases@(E = m * c ^ 2, list@(5 ^ 6, [and so on]), integral@(x) - 3 * d * x, integral@(x * differential-d@(x))) = b" xml:id="S5.Ex41.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMDual>
<XMApp>
<XMTok meaning="cases"/>
<XMRef idref="S5.Ex41.m1.1"/>
<XMRef idref="S5.Ex41.m1.2"/>
<XMRef idref="S5.Ex41.m1.3"/>
<XMRef idref="S5.Ex41.m1.4"/>
</XMApp>
<XMWrap>
<XMArray>
<XMRow>
<XMCell align="left">
<XMApp xml:id="S5.Ex41.m1.1">
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">E</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">m</XMTok>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMCell>
<XMCell align="left">
<XMDual xml:id="S5.Ex41.m1.2">
<XMApp>
<XMTok meaning="list"/>
<XMRef idref="S5.Ex41.m1.2.2"/>
<XMRef idref="S5.Ex41.m1.2.1"/>
</XMApp>
<XMWrap>
<XMApp xml:id="S5.Ex41.m1.2.2">
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMTok meaning="5" role="NUMBER">5</XMTok>
<XMTok fontsize="70%" meaning="6" role="NUMBER">6</XMTok>
</XMApp>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMText xml:id="S5.Ex41.m1.2.1">and so on</XMText>
</XMWrap>
</XMDual>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="left">
<XMApp xml:id="S5.Ex41.m1.3">
<XMTok meaning="minus" role="ADDOP">-</XMTok>
<XMApp>
<XMTok mathstyle="text" meaning="integral" name="int" role="INTOP">∫</XMTok>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok meaning="3" role="NUMBER" rpadding="1.7pt">3</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
</XMApp>
</XMApp>
</XMCell>
<XMCell align="left">
<XMApp xml:id="S5.Ex41.m1.4">
<XMTok mathstyle="text" meaning="integral" name="int" role="INTOP">∫</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN" rpadding="1.7pt">x</XMTok>
<XMApp>
<XMTok font="italic" meaning="differential-d" role="DIFFOP">d</XMTok>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMCell>
</XMRow>
</XMArray>
<XMTok role="CLOSE" stretchy="true">}</XMTok>
</XMWrap>
</XMDual>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
</XMApp>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S5.p5">
<equation xml:id="S5.Ex42">
<Math mode="display" tex="\begin{rcases*}x^{2}&for $\int x\,dx>0$\\ x^{3}&else\end{rcases*}\Rightarrow\cdots" text="cases@(x ^ 2, [for >∫xdx0], x ^ 3, [else]) Rightarrow cdots" xml:id="S5.Ex42.m1">
<XMath>
<XMApp>
<XMTok name="Rightarrow" role="ARROW">⇒</XMTok>
<XMDual>
<XMApp>
<XMTok meaning="cases"/>
<XMRef idref="S5.Ex42.m1.1"/>
<XMRef idref="S5.Ex42.m1.2"/>
<XMRef idref="S5.Ex42.m1.3"/>
<XMRef idref="S5.Ex42.m1.4"/>
</XMApp>
<XMWrap>
<XMArray>
<XMRow>
<XMCell align="left">
<XMApp xml:id="S5.Ex42.m1.1">
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
</XMCell>
<XMCell align="left">
<XMText xml:id="S5.Ex42.m1.2">for <Math mode="inline" tex="\int x\,dx>0" text="integral@(x * differential-d@(x)) > 0" xml:id="S5.Ex42.m1.m1">
<XMath>
<XMApp>
<XMTok meaning="greater-than" role="RELOP">></XMTok>
<XMApp>
<XMTok mathstyle="text" meaning="integral" name="int" role="INTOP">∫</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN" rpadding="1.7pt">x</XMTok>
<XMApp>
<XMTok font="italic" meaning="differential-d" role="DIFFOP">d</XMTok>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
</XMApp>
</XMApp>
</XMApp>
<XMTok meaning="0" role="NUMBER">0</XMTok>
</XMApp>
</XMath>
</Math></XMText>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="left">
<XMApp xml:id="S5.Ex42.m1.3">
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
<XMTok fontsize="70%" meaning="3" role="NUMBER">3</XMTok>
</XMApp>
</XMCell>
<XMCell align="left">
<XMText xml:id="S5.Ex42.m1.4">else</XMText>
</XMCell>
</XMRow>
</XMArray>
<XMTok role="CLOSE" stretchy="true">}</XMTok>
</XMWrap>
</XMDual>
<XMTok name="cdots" role="ID">⋯</XMTok>
</XMApp>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S5.p6">
<equation xml:id="S5.Ex43">
<Math mode="display" tex="\begin{drcases}E=mc^{2}&5^{6}\quad\text{and so on}\\ \int x-3\,dx&\int x\,dx\end{drcases}=b" text="cases@(E = m * c ^ 2, list@(5 ^ 6, [and so on]), integral@(x) - 3 * d * x, integral@(x * differential-d@(x))) = b" xml:id="S5.Ex43.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMDual>
<XMApp>
<XMTok meaning="cases"/>
<XMRef idref="S5.Ex43.m1.1"/>
<XMRef idref="S5.Ex43.m1.2"/>
<XMRef idref="S5.Ex43.m1.3"/>
<XMRef idref="S5.Ex43.m1.4"/>
</XMApp>
<XMWrap>
<XMArray>
<XMRow>
<XMCell align="left">
<XMApp xml:id="S5.Ex43.m1.1">
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">E</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">m</XMTok>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMCell>
<XMCell align="left">
<XMDual xml:id="S5.Ex43.m1.2">
<XMApp>
<XMTok meaning="list"/>
<XMRef idref="S5.Ex43.m1.2.2"/>
<XMRef idref="S5.Ex43.m1.2.1"/>
</XMApp>
<XMWrap>
<XMApp xml:id="S5.Ex43.m1.2.2">
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMTok meaning="5" role="NUMBER">5</XMTok>
<XMTok fontsize="70%" meaning="6" role="NUMBER">6</XMTok>
</XMApp>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMText xml:id="S5.Ex43.m1.2.1">and so on</XMText>
</XMWrap>
</XMDual>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="left">
<XMApp xml:id="S5.Ex43.m1.3">
<XMTok meaning="minus" role="ADDOP">-</XMTok>
<XMApp>
<XMTok mathstyle="display" meaning="integral" name="int" role="INTOP">∫</XMTok>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok meaning="3" role="NUMBER" rpadding="1.7pt">3</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
</XMApp>
</XMApp>
</XMCell>
<XMCell align="left">
<XMApp xml:id="S5.Ex43.m1.4">
<XMTok mathstyle="display" meaning="integral" name="int" role="INTOP">∫</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN" rpadding="1.7pt">x</XMTok>
<XMApp>
<XMTok font="italic" meaning="differential-d" role="DIFFOP">d</XMTok>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMCell>
</XMRow>
</XMArray>
<XMTok role="CLOSE" stretchy="true">}</XMTok>
</XMWrap>
</XMDual>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
</XMApp>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S5.p7">
<equation xml:id="S5.Ex44">
<Math mode="display" tex="\begin{drcases*}x^{2}&for $\int x\,dx>0$\\ \int x^{3}\,x&else\end{drcases*}\Rightarrow\cdots" text="cases@(x ^ 2, [for >∫xdx0], integral@(x ^ 3 * x), [else]) Rightarrow cdots" xml:id="S5.Ex44.m1">
<XMath>
<XMApp>
<XMTok name="Rightarrow" role="ARROW">⇒</XMTok>
<XMDual>
<XMApp>
<XMTok meaning="cases"/>
<XMRef idref="S5.Ex44.m1.1"/>
<XMRef idref="S5.Ex44.m1.2"/>
<XMRef idref="S5.Ex44.m1.3"/>
<XMRef idref="S5.Ex44.m1.4"/>
</XMApp>
<XMWrap>
<XMArray>
<XMRow>
<XMCell align="left">
<XMApp xml:id="S5.Ex44.m1.1">
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
</XMCell>
<XMCell align="left">
<XMText xml:id="S5.Ex44.m1.2">for <Math mode="inline" tex="\int x\,dx>0" text="integral@(x * differential-d@(x)) > 0" xml:id="S5.Ex44.m1.m1">
<XMath>
<XMApp>
<XMTok meaning="greater-than" role="RELOP">></XMTok>
<XMApp>
<XMTok mathstyle="text" meaning="integral" name="int" role="INTOP">∫</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN" rpadding="1.7pt">x</XMTok>
<XMApp>
<XMTok font="italic" meaning="differential-d" role="DIFFOP">d</XMTok>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
</XMApp>
</XMApp>
</XMApp>
<XMTok meaning="0" role="NUMBER">0</XMTok>
</XMApp>
</XMath>
</Math></XMText>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="left">
<XMApp xml:id="S5.Ex44.m1.3">
<XMTok mathstyle="display" meaning="integral" name="int" role="INTOP">∫</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMApp rpadding="1.7pt">
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
<XMTok fontsize="70%" meaning="3" role="NUMBER">3</XMTok>
</XMApp>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
</XMApp>
</XMApp>
</XMCell>
<XMCell align="left">
<XMText xml:id="S5.Ex44.m1.4">else</XMText>
</XMCell>
</XMRow>
</XMArray>
<XMTok role="CLOSE" stretchy="true">}</XMTok>
</XMWrap>
</XMDual>
<XMTok name="cdots" role="ID">⋯</XMTok>
</XMApp>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S5.p8">
<equation xml:id="S5.Ex45">
<Math mode="display" tex="\text{foo}=\begin{cases*}\pi&if something\\ \int\Omega^{\Xi}\,\Omega&otherwise\end{cases*}" text="[foo] = cases@(pi, [if something], integral@(Omega ^ Xi * Omega), [otherwise])" xml:id="S5.Ex45.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMText>foo</XMText>
<XMDual>
<XMApp>
<XMTok meaning="cases"/>
<XMRef idref="S5.Ex45.m1.1"/>
<XMRef idref="S5.Ex45.m1.2"/>
<XMRef idref="S5.Ex45.m1.3"/>
<XMRef idref="S5.Ex45.m1.4"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="true">{</XMTok>
<XMArray>
<XMRow>
<XMCell align="left">
<XMTok font="italic" name="pi" role="UNKNOWN" xml:id="S5.Ex45.m1.1">π</XMTok>
</XMCell>
<XMCell align="left">
<XMText xml:id="S5.Ex45.m1.2">if something</XMText>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="left">
<XMApp xml:id="S5.Ex45.m1.3">
<XMTok mathstyle="text" meaning="integral" name="int" role="INTOP">∫</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMApp rpadding="1.7pt">
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMTok name="Omega" role="UNKNOWN">Ω</XMTok>
<XMTok fontsize="70%" name="Xi" role="UNKNOWN">Ξ</XMTok>
</XMApp>
<XMTok name="Omega" role="UNKNOWN">Ω</XMTok>
</XMApp>
</XMApp>
</XMCell>
<XMCell align="left">
<XMText xml:id="S5.Ex45.m1.4">otherwise</XMText>
</XMCell>
</XMRow>
</XMArray>
</XMWrap>
</XMDual>
</XMApp>
</XMath>
</Math>
</equation>
</para>
</section>
<section inlist="toc" xml:id="S6">
<tags>
<tag>6</tag>
<tag role="refnum">6</tag>
<tag role="typerefnum">§6</tag>
</tags>
<title><tag close=" ">6</tag>Gathered</title>
<para xml:id="S6.p1">
<equation xml:id="S6.Ex46">
<Math mode="display" tex="A=\begin{gathered}\framebox[113.81102pt]{first}\\ \framebox[113.81102pt]{last}\end{gathered}B" text="A = [first] * [last] * B" xml:id="S6.Ex46.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">A</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMDual>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S6.Ex46.m1.1"/>
<XMRef idref="S6.Ex46.m1.2"/>
</XMApp>
<XMArray name="gathered" vattach="top">
<XMRow>
<XMCell align="center">
<XMText framecolor="#000000" framed="rectangle" width="113.8pt" xml:id="S6.Ex46.m1.1">first</XMText>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="center">
<XMText framecolor="#000000" framed="rectangle" width="113.8pt" xml:id="S6.Ex46.m1.2">last</XMText>
</XMCell>
</XMRow>
</XMArray>
</XMDual>
<XMTok font="italic" role="UNKNOWN">B</XMTok>
</XMApp>
</XMApp>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S6.p2">
<equation xml:id="S6.Ex47">
<Math mode="display" tex="\begin{gathered}a=b+c\\ b=c+d\\ ...\end{gathered}" text="a = b + c * b = c + d * ldots" xml:id="S6.Ex47.m1">
<XMath>
<XMDual>
<XMApp>
<XMTok meaning="multirelation"/>
<XMRef idref="S6.Ex47.m1.1"/>
<XMRef idref="S6.Ex47.m1.2"/>
<XMApp>
<XMRef idref="S6.Ex47.m1.4"/>
<XMRef idref="S6.Ex47.m1.3"/>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S6.Ex47.m1.5"/>
<XMRef idref="S6.Ex47.m1.6"/>
</XMApp>
</XMApp>
<XMRef idref="S6.Ex47.m1.7"/>
<XMApp>
<XMRef idref="S6.Ex47.m1.9"/>
<XMRef idref="S6.Ex47.m1.8"/>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S6.Ex47.m1.10"/>
<XMRef idref="S6.Ex47.m1.11"/>
</XMApp>
</XMApp>
</XMApp>
<XMArray name="gathered" vattach="middle">
<XMRow>
<XMCell align="center">
<XMApp>
<XMTok meaning="equals" role="RELOP" xml:id="S6.Ex47.m1.2">=</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S6.Ex47.m1.1">a</XMTok>
<XMApp>
<XMTok meaning="plus" role="ADDOP" xml:id="S6.Ex47.m1.4">+</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S6.Ex47.m1.3">b</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S6.Ex47.m1.5">c</XMTok>
</XMApp>
</XMApp>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="center">
<XMApp>
<XMTok meaning="equals" role="RELOP" xml:id="S6.Ex47.m1.7">=</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S6.Ex47.m1.6">b</XMTok>
<XMApp>
<XMTok meaning="plus" role="ADDOP" xml:id="S6.Ex47.m1.9">+</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S6.Ex47.m1.8">c</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S6.Ex47.m1.10">d</XMTok>
</XMApp>
</XMApp>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="center">
<XMTok name="ldots" role="ID" xml:id="S6.Ex47.m1.11">…</XMTok>
</XMCell>
</XMRow>
</XMArray>
</XMDual>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S6.p3">
<equation xml:id="S6.Ex48">
<Math mode="display" tex="\boxed{hello}" text="h * e * l * l * o" xml:id="S6.Ex48.m1">
<XMath>
<XMApp enclose="box">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">h</XMTok>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
<XMTok font="italic" role="UNKNOWN">l</XMTok>
<XMTok font="italic" role="UNKNOWN">l</XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
</XMApp>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S6.p4">
<equationgroup class="ltx_eqn_align" xml:id="S12.EGx3">
<equation xml:id="S6.Ex49">
<MathFork>
<Math tex="\displaystyle f(x)=\int h(x)\,dx" text="f * x = integral@(h * x * differential-d@(x))" xml:id="S6.Ex49.m3">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">f</XMTok>
<XMDual>
<XMRef idref="S6.Ex49.m3.1"/>
<XMWrap>
<XMTok role="OPEN" stretchy="false">(</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S6.Ex49.m3.1">x</XMTok>
<XMTok role="CLOSE" stretchy="false">)</XMTok>
</XMWrap>
</XMDual>
</XMApp>
<XMApp>
<XMTok mathstyle="display" meaning="integral" name="int" role="INTOP">∫</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">h</XMTok>
<XMDual>
<XMRef idref="S6.Ex49.m3.2"/>
<XMWrap>
<XMTok role="OPEN" stretchy="false">(</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S6.Ex49.m3.2">x</XMTok>
<XMTok role="CLOSE" rpadding="1.7pt" stretchy="false">)</XMTok>
</XMWrap>
</XMDual>
<XMApp>
<XMTok font="italic" meaning="differential-d" role="DIFFOP">d</XMTok>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMApp>
</XMath>
</Math>
<MathBranch>
<td align="right"><Math mode="inline" tex="\displaystyle f(x)" text="f * x" xml:id="S6.Ex49.m1">
<XMath>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">f</XMTok>
<XMDual>
<XMRef idref="S6.Ex49.m1.1"/>
<XMWrap>
<XMTok role="OPEN" stretchy="false">(</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S6.Ex49.m1.1">x</XMTok>
<XMTok role="CLOSE" stretchy="false">)</XMTok>
</XMWrap>
</XMDual>
</XMApp>
</XMath>
</Math></td>
<td align="left"><Math mode="inline" tex="\displaystyle=\int h(x)\,dx" text="absent = integral@(h * x * differential-d@(x))" xml:id="S6.Ex49.m2">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok meaning="absent"/>
<XMApp>
<XMTok mathstyle="display" meaning="integral" name="int" role="INTOP">∫</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">h</XMTok>
<XMDual>
<XMRef idref="S6.Ex49.m2.1"/>
<XMWrap>
<XMTok role="OPEN" stretchy="false">(</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S6.Ex49.m2.1">x</XMTok>
<XMTok role="CLOSE" rpadding="1.7pt" stretchy="false">)</XMTok>
</XMWrap>
</XMDual>
<XMApp>
<XMTok font="italic" meaning="differential-d" role="DIFFOP">d</XMTok>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMApp>
</XMath>
</Math></td>
</MathBranch>
</MathFork>
</equation>
<equation xml:id="S6.Ex50">
<MathFork>
<Math tex="\displaystyle=g(x)" text="absent = g * x" xml:id="S6.Ex50.m2">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok meaning="absent"/>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">g</XMTok>
<XMDual>
<XMRef idref="S6.Ex50.m2.1"/>
<XMWrap>
<XMTok role="OPEN" stretchy="false">(</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S6.Ex50.m2.1">x</XMTok>
<XMTok role="CLOSE" stretchy="false">)</XMTok>
</XMWrap>
</XMDual>
</XMApp>
</XMApp>
</XMath>
</Math>
<MathBranch>
<td/>
<td align="left"><Math mode="inline" tex="\displaystyle=g(x)" text="absent = g * x" xml:id="S6.Ex50.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok meaning="absent"/>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">g</XMTok>
<XMDual>
<XMRef idref="S6.Ex50.m1.1"/>
<XMWrap>
<XMTok role="OPEN" stretchy="false">(</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S6.Ex50.m1.1">x</XMTok>
<XMTok role="CLOSE" stretchy="false">)</XMTok>
</XMWrap>
</XMDual>
</XMApp>
</XMApp>
</XMath>
</Math></td>
</MathBranch>
</MathFork>
</equation>
</equationgroup>
</para>
<para xml:id="S6.p5">
<equationgroup class="ltx_eqn_align" xml:id="S12.EGx4">
<equation xml:id="S6.E11">
<tags>
<tag>(11)</tag>
<tag role="refnum">11</tag>
</tags>
<MathFork>
<Math tex="\displaystyle a=b" text="a = b" xml:id="S6.E11.m3">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
</XMApp>
</XMath>
</Math>
<MathBranch>
<td align="right"><Math mode="inline" tex="\displaystyle a" text="a" xml:id="S6.E11.m1">
<XMath>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
</XMath>
</Math></td>
<td align="left"><Math mode="inline" tex="\displaystyle=b" text="absent = b" xml:id="S6.E11.m2">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok meaning="absent"/>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
</XMApp>
</XMath>
</Math></td>
</MathBranch>
</MathFork>
</equation>
<p class="ltx_intertext">Some text</p>
<equation xml:id="S6.E12">
<tags>
<tag>(12)</tag>
<tag role="refnum">12</tag>
</tags>
<MathFork>
<Math tex="\displaystyle c=d" text="c = d" xml:id="S6.E12.m3">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
</XMApp>
</XMath>
</Math>
<MathBranch>
<td align="right"><Math mode="inline" tex="\displaystyle c" text="c" xml:id="S6.E12.m1">
<XMath>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
</XMath>
</Math></td>
<td align="left"><Math mode="inline" tex="\displaystyle=d" text="absent = d" xml:id="S6.E12.m2">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok meaning="absent"/>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
</XMApp>
</XMath>
</Math></td>
</MathBranch>
</MathFork>
</equation>
<p class="ltx_intertext">Some short text</p>
<equation xml:id="S6.E13">
<tags>
<tag>(13)</tag>
<tag role="refnum">13</tag>
</tags>
<MathFork>
<Math tex="\displaystyle e=f" text="e = f" xml:id="S6.E13.m3">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
<XMTok font="italic" role="UNKNOWN">f</XMTok>
</XMApp>
</XMath>
</Math>
<MathBranch>
<td align="right"><Math mode="inline" tex="\displaystyle e" text="e" xml:id="S6.E13.m1">
<XMath>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
</XMath>
</Math></td>
<td align="left"><Math mode="inline" tex="\displaystyle=f" text="absent = f" xml:id="S6.E13.m2">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok meaning="absent"/>
<XMTok font="italic" role="UNKNOWN">f</XMTok>
</XMApp>
</XMath>
</Math></td>
</MathBranch>
</MathFork>
</equation>
</equationgroup>
</para>
</section>
<section inlist="toc" xml:id="S7">
<tags>
<tag>7</tag>
<tag role="refnum">7</tag>
<tag role="typerefnum">§7</tag>
</tags>
<title><tag close=" ">7</tag>Delimiters</title>
<para xml:id="S7.p1">
<equation xml:id="S7.Ex51">
<Math mode="display" tex="\lvert\frac{a}{c}\rvert\quad\left\lvert\frac{a}{c}\right\rvert\quad\Bigg{% \lvert}\frac{a}{b}\Bigg{\rvert}" text="list@(absolute-value@(a / c), absolute-value@(a / c), absolute-value@(a / b))" xml:id="S7.Ex51.m1">
<XMath>
<XMDual>
<XMApp>
<XMTok meaning="list"/>
<XMRef idref="S7.Ex51.m1.4"/>
<XMRef idref="S7.Ex51.m1.5"/>
<XMRef idref="S7.Ex51.m1.6"/>
</XMApp>
<XMWrap>
<XMDual xml:id="S7.Ex51.m1.4">
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S7.Ex51.m1.1"/>
</XMApp>
<XMWrap>
<XMTok name="lvert" role="OPEN" stretchy="false">|</XMTok>
<XMApp xml:id="S7.Ex51.m1.1">
<XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
</XMApp>
<XMTok name="rvert" role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMDual xml:id="S7.Ex51.m1.5">
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S7.Ex51.m1.2"/>
</XMApp>
<XMWrap>
<XMTok name="lvert" role="OPEN" stretchy="true">|</XMTok>
<XMApp xml:id="S7.Ex51.m1.2">
<XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
</XMApp>
<XMTok name="rvert" role="CLOSE" stretchy="true">|</XMTok>
</XMWrap>
</XMDual>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMDual xml:id="S7.Ex51.m1.6">
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S7.Ex51.m1.3"/>
</XMApp>
<XMWrap>
<XMTok fontsize="260%" name="lvert" role="OPEN" stretchy="false">|</XMTok>
<XMApp xml:id="S7.Ex51.m1.3">
<XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
</XMApp>
<XMTok fontsize="260%" name="rvert" role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
</XMWrap>
</XMDual>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S7.p2">
<equation xml:id="S7.Ex52">
<Math mode="display" tex="\lvert\frac{a}{b}\rvert\quad\left\lvert\frac{a}{b}\right\rvert\quad\big{\lvert% }\frac{a}{b}\big{\rvert}\quad\Bigg{\lvert}\frac{a}{b}\Bigg{\rvert}" text="list@(absolute-value@(a / b), absolute-value@(a / b), absolute-value@(a / b), absolute-value@(a / b))" xml:id="S7.Ex52.m1">
<XMath>
<XMDual>
<XMApp>
<XMTok meaning="list"/>
<XMRef idref="S7.Ex52.m1.5"/>
<XMRef idref="S7.Ex52.m1.6"/>
<XMRef idref="S7.Ex52.m1.7"/>
<XMRef idref="S7.Ex52.m1.8"/>
</XMApp>
<XMWrap>
<XMDual xml:id="S7.Ex52.m1.5">
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S7.Ex52.m1.1"/>
</XMApp>
<XMWrap>
<XMTok name="lvert" role="OPEN" stretchy="false">|</XMTok>
<XMApp xml:id="S7.Ex52.m1.1">
<XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
</XMApp>
<XMTok name="rvert" role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMDual xml:id="S7.Ex52.m1.6">
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S7.Ex52.m1.2"/>
</XMApp>
<XMWrap>
<XMTok name="lvert" role="OPEN" stretchy="true">|</XMTok>
<XMApp xml:id="S7.Ex52.m1.2">
<XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
</XMApp>
<XMTok name="rvert" role="CLOSE" stretchy="true">|</XMTok>
</XMWrap>
</XMDual>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMDual xml:id="S7.Ex52.m1.7">
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S7.Ex52.m1.3"/>
</XMApp>
<XMWrap>
<XMTok fontsize="120%" name="lvert" role="OPEN" stretchy="false">|</XMTok>
<XMApp xml:id="S7.Ex52.m1.3">
<XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
</XMApp>
<XMTok fontsize="120%" name="rvert" role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMDual xml:id="S7.Ex52.m1.8">
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S7.Ex52.m1.4"/>
</XMApp>
<XMWrap>
<XMTok fontsize="260%" name="lvert" role="OPEN" stretchy="false">|</XMTok>
<XMApp xml:id="S7.Ex52.m1.4">
<XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
</XMApp>
<XMTok fontsize="260%" name="rvert" role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
</XMWrap>
</XMDual>
</XMath>
</Math>
</equation>
<equation xml:id="S7.Ex53">
<Math mode="display" tex="\lvert@\pi@\rvert\quad\left\lvert\_\phi\_\right\rvert" text="list@(absolute-value@(@ * pi * @), absolute-value@(_ * phi * _))" xml:id="S7.Ex53.m1">
<XMath>
<XMDual>
<XMApp>
<XMTok meaning="list"/>
<XMRef idref="S7.Ex53.m1.1"/>
<XMRef idref="S7.Ex53.m1.2"/>
</XMApp>
<XMWrap>
<XMDual xml:id="S7.Ex53.m1.1">
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S7.Ex53.m1.1.1"/>
</XMApp>
<XMWrap>
<XMTok name="lvert" role="OPEN" stretchy="false">|</XMTok>
<XMApp xml:id="S7.Ex53.m1.1.1">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok role="UNKNOWN">@</XMTok>
<XMTok font="italic" name="pi" role="UNKNOWN">π</XMTok>
<XMTok role="UNKNOWN">@</XMTok>
</XMApp>
<XMTok name="rvert" role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMDual xml:id="S7.Ex53.m1.2">
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S7.Ex53.m1.2.1"/>
</XMApp>
<XMWrap>
<XMTok name="lvert" role="OPEN" stretchy="true">|</XMTok>
<XMApp xml:id="S7.Ex53.m1.2.1">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok role="UNKNOWN">_</XMTok>
<XMTok font="italic" name="phi" role="UNKNOWN">ϕ</XMTok>
<XMTok role="UNKNOWN">_</XMTok>
</XMApp>
<XMTok name="rvert" role="CLOSE" stretchy="true">|</XMTok>
</XMWrap>
</XMDual>
</XMWrap>
</XMDual>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S7.p3">
<equation xml:id="S7.Ex54">
<Math mode="display" tex="\left\langle A,\frac{1}{2}\right\rangle\quad\Big{\langle}B\,\Big{|}\,\sum_{k}f% _{k}\,\Big{|}\,C\Big{\rangle}" text="list@(list@(A, 1 / 2), quantum-operator-product@(B, (sum _ k)@(f _ k), C))" xml:id="S7.Ex54.m1">
<XMath>
<XMDual>
<XMApp>
<XMTok meaning="list"/>
<XMRef idref="S7.Ex54.m1.5"/>
<XMRef idref="S7.Ex54.m1.6"/>
</XMApp>
<XMWrap>
<XMDual xml:id="S7.Ex54.m1.5">
<XMApp>
<XMTok meaning="list"/>
<XMRef idref="S7.Ex54.m1.1"/>
<XMRef idref="S7.Ex54.m1.2"/>
</XMApp>
<XMWrap>
<XMTok name="langle" role="OPEN" stretchy="true">⟨</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S7.Ex54.m1.1">A</XMTok>
<XMTok role="PUNCT">,</XMTok>
<XMApp xml:id="S7.Ex54.m1.2">
<XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
<XMTok meaning="1" role="NUMBER">1</XMTok>
<XMTok meaning="2" role="NUMBER">2</XMTok>
</XMApp>
<XMTok name="rangle" role="CLOSE" stretchy="true">⟩</XMTok>
</XMWrap>
</XMDual>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMDual xml:id="S7.Ex54.m1.6">
<XMApp>
<XMTok meaning="quantum-operator-product"/>
<XMRef idref="S7.Ex54.m1.4"/>
<XMRef idref="S7.Ex54.m1.6.1"/>
<XMRef idref="S7.Ex54.m1.3"/>
</XMApp>
<XMWrap>
<XMTok fontsize="160%" name="langle" role="OPEN" stretchy="false">⟨</XMTok>
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<XMTok fontsize="160%" role="CLOSE" rpadding="1.7pt" stretchy="false">|</XMTok>
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<para xml:id="S7.p4">
<equation xml:id="S7.Ex55">
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<XMTok font="italic" role="UNKNOWN">X</XMTok>
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<XMApp>
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<XMTok meaning="3" role="NUMBER" xml:id="S7.p5.m2.3"> 3</XMTok>
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<Math mode="inline" tex="\big{\langle}1\,\big{|}\,\frac{8}{\frac{4}{1}}\,\big{|}\,3\big{\rangle}" text="quantum-operator-product@(1, 8 / 4 / 1, 3)" xml:id="S7.p5.m3">
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<XMTok fontsize="50%" meaning="1" role="NUMBER">1</XMTok>
</XMApp>
</XMApp>
<XMTok fontsize="120%" role="OPEN" stretchy="false">|</XMTok>
<XMTok meaning="3" role="NUMBER" xml:id="S7.p5.m3.1"> 3</XMTok>
<XMTok fontsize="120%" name="rangle" role="CLOSE" stretchy="false">⟩</XMTok>
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</XMath>
</Math></p>
</para>
<para xml:id="S7.p6">
<equation xml:id="S7.Ex56">
<Math mode="display" tex="\left(\frac{\pi}{\omega}\right)\cdot\left[\int xdx\right]\ldots[\sqrt{\frac{% \sin x}{\cos z}}]\cdots\lparen\frac{\frac{foo}{bar}}{\frac{baz}{qux}}\rparen" text="((pi / omega) cdot delimited-[]@(integral@(x * differential-d@(x)))) * ldots * delimited-[]@(square-root@(sine@(x) / cosine@(z))) * cdots * ((f * o * o) / (b * a * r) / (b * a * z) / (q * u * x))" xml:id="S7.Ex56.m1">
<XMath>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMApp>
<XMTok name="cdot" role="MULOP">⋅</XMTok>
<XMDual>
<XMRef idref="S7.Ex56.m1.1"/>
<XMWrap>
<XMTok role="OPEN" stretchy="true">(</XMTok>
<XMApp xml:id="S7.Ex56.m1.1">
<XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
<XMTok font="italic" name="pi" role="UNKNOWN">π</XMTok>
<XMTok font="italic" name="omega" role="UNKNOWN">ω</XMTok>
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<XMTok role="CLOSE" stretchy="true">)</XMTok>
</XMWrap>
</XMDual>
<XMDual>
<XMApp>
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<XMTok role="OPEN" stretchy="true">[</XMTok>
<XMApp xml:id="S7.Ex56.m1.4">
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<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
<XMApp>
<XMTok font="italic" meaning="differential-d" role="DIFFOP">d</XMTok>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
</XMApp>
</XMApp>
</XMApp>
<XMTok role="CLOSE" stretchy="true">]</XMTok>
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</XMDual>
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<XMTok name="ldots" role="ID">…</XMTok>
<XMDual>
<XMApp>
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<XMRef idref="S7.Ex56.m1.2"/>
</XMApp>
<XMWrap>
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<XMApp xml:id="S7.Ex56.m1.2">
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<XMApp>
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<XMApp>
<XMTok meaning="sine" role="TRIGFUNCTION">sin</XMTok>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
</XMApp>
<XMApp>
<XMTok meaning="cosine" role="TRIGFUNCTION">cos</XMTok>
<XMTok font="italic" role="UNKNOWN">z</XMTok>
</XMApp>
</XMApp>
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<XMTok role="CLOSE" stretchy="false">]</XMTok>
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<XMTok name="cdots" role="ID">⋯</XMTok>
<XMDual>
<XMRef idref="S7.Ex56.m1.3"/>
<XMWrap>
<XMTok name="lparen" role="OPEN" stretchy="false">(</XMTok>
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<XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
<XMApp>
<XMTok mathstyle="text" meaning="divide" role="FRACOP"/>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">f</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">o</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">o</XMTok>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">b</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">a</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">r</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok mathstyle="text" meaning="divide" role="FRACOP"/>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">b</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">a</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">z</XMTok>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">q</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">u</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">x</XMTok>
</XMApp>
</XMApp>
</XMApp>
<XMTok name="rparen" role="CLOSE" stretchy="false">)</XMTok>
</XMWrap>
</XMDual>
</XMApp>
</XMath>
</Math>
</equation>
</para>
<paragraph inlist="toc" xml:id="S7.SS0.SSS0.Px1">
<title>Operators</title>
<para xml:id="S7.SS0.SSS0.Px1.p1">
<equation xml:id="S7.Ex57">
<Math mode="display" tex="a:=b\quad a\vcentcolon=b\quad a\ordinarycolon=b" text="formulae@(a assign b, formulae@(a assign b, a assign b))" xml:id="S7.Ex57.m1">
<XMath>
<XMDual>
<XMApp>
<XMTok meaning="formulae"/>
<XMRef idref="S7.Ex57.m1.1"/>
<XMRef idref="S7.Ex57.m1.2"/>
</XMApp>
<XMWrap>
<XMApp xml:id="S7.Ex57.m1.1">
<XMTok meaning="assign" role="RELOP">:=</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
</XMApp>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMDual xml:id="S7.Ex57.m1.2">
<XMApp>
<XMTok meaning="formulae"/>
<XMRef idref="S7.Ex57.m1.2.1"/>
<XMRef idref="S7.Ex57.m1.2.2"/>
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<XMTok meaning="assign" role="RELOP">:=</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
</XMApp>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMApp xml:id="S7.Ex57.m1.2.2">
<XMTok meaning="assign" role="RELOP">:=</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
</XMApp>
</XMWrap>
</XMDual>
</XMWrap>
</XMDual>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S7.SS0.SSS0.Px1.p2">
<equation xml:id="S7.Ex58">
<Math mode="display" tex="a\coloneqq b\quad c\Colonapprox d\quad e\dblcolon f" text="formulae@(a coloneqq b, formulae@(c Colonapprox d, e dblcolon f))" xml:id="S7.Ex58.m1">
<XMath>
<XMDual>
<XMApp>
<XMTok meaning="formulae"/>
<XMRef idref="S7.Ex58.m1.1"/>
<XMRef idref="S7.Ex58.m1.2"/>
</XMApp>
<XMWrap>
<XMApp xml:id="S7.Ex58.m1.1">
<XMTok name="coloneqq" role="RELOP">≔</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
</XMApp>
<XMHint name="quad" role="PUNCT" width="10pt"/>
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<XMApp>
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<XMRef idref="S7.Ex58.m1.2.2"/>
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<XMApp xml:id="S7.Ex58.m1.2.1">
<XMTok name="Colonapprox" role="RELOP">::≈</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
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<XMApp xml:id="S7.Ex58.m1.2.2">
<XMTok name="dblcolon" role="RELOP">::</XMTok>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
<XMTok font="italic" role="UNKNOWN">f</XMTok>
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</XMWrap>
</XMDual>
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</equation>
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<para xml:id="S7.SS0.SSS0.Px1.p3">
<equation xml:id="S7.Ex59">
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<XMath>
<XMDual>
<XMApp>
<XMTok meaning="list"/>
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<XMRef idref="S7.Ex59.m1.3"/>
<XMRef idref="S7.Ex59.m1.4"/>
<XMRef idref="S7.Ex59.m1.5"/>
<XMRef idref="S7.Ex59.m1.6"/>
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<XMWrap>
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<XMTok role="PUNCT"></XMTok>
<XMTok meaning="times" role="MULOP" xml:id="S7.Ex59.m1.2">×</XMTok>
<XMTok role="PUNCT"></XMTok>
<XMTok name="nuparrow" role="ARROW" xml:id="S7.Ex59.m1.3">⤉</XMTok>
<XMTok role="PUNCT"></XMTok>
<XMTok name="ndownarrow" role="ARROW" xml:id="S7.Ex59.m1.4">⤈</XMTok>
<XMTok role="PUNCT"></XMTok>
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<XMTok role="PUNCT"></XMTok>
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</Math>
</equation>
</para>
</paragraph>
</section>
<section inlist="toc" xml:id="S8">
<tags>
<tag>8</tag>
<tag role="refnum">8</tag>
<tag role="typerefnum">§8</tag>
</tags>
<title><tag close=" ">8</tag>Prescripts</title>
<para xml:id="S8.p1">
<equation xml:id="S8.Ex60">
<Math mode="display" tex="{}^{4}_{12}\mathbf{C}^{5+}_{2}\quad\prescript{14}{2}{\mathbf{C}}^{5+}_{2}\quad% \prescript{4}{12}{\mathbf{C}}^{5+}_{2}\quad\prescript{14}{}{\mathbf{C}}^{5+}_{% 2}\quad\prescript{}{2}{\mathbf{C}}^{5+}_{2}" text="list@(^ 4 _ 12 (C ^ (limit-from@(5, +))) _ 2, ^ 14 _ 2 (C ^ (limit-from@(5, +))) _ 2, ^ 4 _ 12 (C ^ (limit-from@(5, +))) _ 2, ^ 14 (C ^ (limit-from@(5, +))) _ 2, _ 2 (C ^ (limit-from@(5, +))) _ 2)" xml:id="S8.Ex60.m1">
<XMath>
<XMDual>
<XMApp>
<XMTok meaning="list"/>
<XMRef idref="S8.Ex60.m1.1"/>
<XMRef idref="S8.Ex60.m1.2"/>
<XMRef idref="S8.Ex60.m1.3"/>
<XMRef idref="S8.Ex60.m1.4"/>
<XMRef idref="S8.Ex60.m1.5"/>
</XMApp>
<XMWrap>
<XMApp xml:id="S8.Ex60.m1.1">
<XMTok role="SUPERSCRIPTOP" scriptpos="pre1"/>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="pre1"/>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
<XMTok font="bold" role="UNKNOWN">C</XMTok>
<XMApp>
<XMTok meaning="limit-from"/>
<XMTok fontsize="70%" meaning="5" role="NUMBER">5</XMTok>
<XMTok fontsize="70%" meaning="plus" role="ADDOP">+</XMTok>
</XMApp>
</XMApp>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
<XMTok fontsize="70%" meaning="12" role="NUMBER">12</XMTok>
</XMApp>
<XMTok fontsize="70%" meaning="4" role="NUMBER">4</XMTok>
</XMApp>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMApp xml:id="S8.Ex60.m1.2">
<XMTok role="SUPERSCRIPTOP" scriptpos="pre1"/>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="pre1"/>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
<XMTok font="bold" role="UNKNOWN">C</XMTok>
<XMApp>
<XMTok meaning="limit-from"/>
<XMTok fontsize="70%" meaning="5" role="NUMBER">5</XMTok>
<XMTok fontsize="70%" meaning="plus" role="ADDOP">+</XMTok>
</XMApp>
</XMApp>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
<XMTok fontsize="70%" meaning="14" role="NUMBER">14</XMTok>
</XMApp>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMApp xml:id="S8.Ex60.m1.3">
<XMTok role="SUPERSCRIPTOP" scriptpos="pre1"/>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="pre1"/>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
<XMTok font="bold" role="UNKNOWN">C</XMTok>
<XMApp>
<XMTok meaning="limit-from"/>
<XMTok fontsize="70%" meaning="5" role="NUMBER">5</XMTok>
<XMTok fontsize="70%" meaning="plus" role="ADDOP">+</XMTok>
</XMApp>
</XMApp>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
<XMTok fontsize="70%" meaning="12" role="NUMBER">12</XMTok>
</XMApp>
<XMTok fontsize="70%" meaning="4" role="NUMBER">4</XMTok>
</XMApp>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMApp xml:id="S8.Ex60.m1.4">
<XMTok role="SUPERSCRIPTOP" scriptpos="pre1"/>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
<XMTok font="bold" role="UNKNOWN">C</XMTok>
<XMApp>
<XMTok meaning="limit-from"/>
<XMTok fontsize="70%" meaning="5" role="NUMBER">5</XMTok>
<XMTok fontsize="70%" meaning="plus" role="ADDOP">+</XMTok>
</XMApp>
</XMApp>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
<XMTok fontsize="70%" meaning="14" role="NUMBER">14</XMTok>
</XMApp>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMApp xml:id="S8.Ex60.m1.5">
<XMTok role="SUBSCRIPTOP" scriptpos="pre1"/>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
<XMTok font="bold" role="UNKNOWN">C</XMTok>
<XMApp>
<XMTok meaning="limit-from"/>
<XMTok fontsize="70%" meaning="5" role="NUMBER">5</XMTok>
<XMTok fontsize="70%" meaning="plus" role="ADDOP">+</XMTok>
</XMApp>
</XMApp>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
</XMWrap>
</XMDual>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S8.p2">
<equation xml:id="S8.Ex61">
<Math mode="display" tex="\prescript{A}{Z}{X}\to\prescript{A-4}{Z-2}{Y}+\prescript{4}{2}{\alpha}" text="^ A _ Z X to ^ A - 4 _ Z - 2 Y + ^ 4 _ 2 alpha" xml:id="S8.Ex61.m1">
<XMath>
<XMApp>
<XMTok name="to" role="ARROW">→</XMTok>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="pre1"/>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="pre1"/>
<XMTok role="UNKNOWN">X</XMTok>
<XMTok font="bold" fontsize="70%" role="UNKNOWN">Z</XMTok>
</XMApp>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">A</XMTok>
</XMApp>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="pre1"/>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="pre1"/>
<XMTok role="UNKNOWN">Y</XMTok>
<XMApp>
<XMTok fontsize="70%" meaning="minus" role="ADDOP">-</XMTok>
<XMTok font="bold" fontsize="70%" role="UNKNOWN">Z</XMTok>
<XMTok font="bold" fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok fontsize="70%" meaning="minus" role="ADDOP">-</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN">A</XMTok>
<XMTok font="italic" fontsize="70%" meaning="4" role="NUMBER">4</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="pre1"/>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="pre1"/>
<XMTok font="italic" name="alpha" role="UNKNOWN">α</XMTok>
<XMTok font="bold" fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
<XMTok font="italic" fontsize="70%" meaning="4" role="NUMBER">4</XMTok>
</XMApp>
</XMApp>
</XMApp>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S8.p3">
<equation xml:id="S8.Ex62">
<Math mode="display" tex="a=\frac{\begin{multlined}xy+xy+\int xy\,\text{dx}+xy+xy\\ +xy+xy+xy+xy\end{multlined}xy+xy+\int xy\,\text{dx}+xy+xy\\ +xy+xy+xy+xy}{z}=\frac{\displaystyle\begin{multlined}xy+xy+\int xy\,\text{dx}+% xy+xy\\ +xy+xy+xy+xy\end{multlined}xy+xy+\int xy\,\text{dx}+xy+xy\\ +xy+xy+xy+xy}{z}" text="a = (x * y + x * y + integral@(x * y * [dx]) + x * y + x * y + x * y + x * y + x * y + x * y) / z = (x * y + x * y + integral@(x * y * [dx]) + x * y + x * y + x * y + x * y + x * y + x * y) / z" xml:id="S8.Ex62.m1">
<XMath>
<XMApp>
<XMTok meaning="multirelation"/>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMApp>
<XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
<XMDual>
<XMApp>
<XMRef idref="S8.Ex62.m1.3"/>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S8.Ex62.m1.1"/>
<XMRef idref="S8.Ex62.m1.2"/>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S8.Ex62.m1.4"/>
<XMRef idref="S8.Ex62.m1.5"/>
</XMApp>
<XMApp>
<XMRef idref="S8.Ex62.m1.7"/>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S8.Ex62.m1.8"/>
<XMRef idref="S8.Ex62.m1.9" rpadding="1.7pt"/>
<XMRef idref="S8.Ex62.m1.10"/>
</XMApp>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S8.Ex62.m1.12"/>
<XMRef idref="S8.Ex62.m1.13"/>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S8.Ex62.m1.15"/>
<XMRef idref="S8.Ex62.m1.16"/>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S8.Ex62.m1.18"/>
<XMRef idref="S8.Ex62.m1.19"/>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S8.Ex62.m1.21"/>
<XMRef idref="S8.Ex62.m1.22"/>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S8.Ex62.m1.24"/>
<XMRef idref="S8.Ex62.m1.25"/>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S8.Ex62.m1.27"/>
<XMRef idref="S8.Ex62.m1.28"/>
</XMApp>
</XMApp>
<XMArray name="splitfrac">
<XMRow>
<XMCell align="left">
<XMApp>
<XMTok meaning="plus" role="ADDOP" xml:id="S8.Ex62.m1.3">+</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.1">x</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.2">y</XMTok>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.4">x</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.5">y</XMTok>
</XMApp>
<XMApp>
<XMTok mathstyle="display" meaning="integral" name="int" role="INTOP" xml:id="S8.Ex62.m1.7">∫</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.8">x</XMTok>
<XMTok font="italic" role="UNKNOWN" rpadding="1.7pt" xml:id="S8.Ex62.m1.9">y</XMTok>
<XMText xml:id="S8.Ex62.m1.10">dx</XMText>
</XMApp>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.12">x</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.13">y</XMTok>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.15">x</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.16">y</XMTok>
</XMApp>
</XMApp>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="right">
<XMApp>
<XMTok meaning="plus" role="ADDOP" xml:id="S8.Ex62.m1.20">+</XMTok>
<XMApp>
<XMTok meaning="plus" role="ADDOP" xml:id="S8.Ex62.m1.17">+</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.18">x</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.19">y</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.21">x</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.22">y</XMTok>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.24">x</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.25">y</XMTok>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.27">x</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.28">y</XMTok>
</XMApp>
</XMApp>
</XMCell>
</XMRow>
</XMArray>
</XMDual>
<XMTok font="italic" role="UNKNOWN">z</XMTok>
</XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMApp>
<XMTok mathstyle="display" meaning="divide" role="FRACOP"/>
<XMDual>
<XMApp>
<XMRef idref="S8.Ex62.m1.31"/>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S8.Ex62.m1.29"/>
<XMRef idref="S8.Ex62.m1.30"/>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S8.Ex62.m1.32"/>
<XMRef idref="S8.Ex62.m1.33"/>
</XMApp>
<XMApp>
<XMRef idref="S8.Ex62.m1.35"/>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S8.Ex62.m1.36"/>
<XMRef idref="S8.Ex62.m1.37" rpadding="1.7pt"/>
<XMRef idref="S8.Ex62.m1.38"/>
</XMApp>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S8.Ex62.m1.40"/>
<XMRef idref="S8.Ex62.m1.41"/>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S8.Ex62.m1.43"/>
<XMRef idref="S8.Ex62.m1.44"/>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S8.Ex62.m1.46"/>
<XMRef idref="S8.Ex62.m1.47"/>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S8.Ex62.m1.49"/>
<XMRef idref="S8.Ex62.m1.50"/>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S8.Ex62.m1.52"/>
<XMRef idref="S8.Ex62.m1.53"/>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S8.Ex62.m1.55"/>
<XMRef idref="S8.Ex62.m1.56"/>
</XMApp>
</XMApp>
<XMArray name="splitdfrac">
<XMRow>
<XMCell align="left">
<XMApp>
<XMTok meaning="plus" role="ADDOP" xml:id="S8.Ex62.m1.31">+</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.29">x</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.30">y</XMTok>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.32">x</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.33">y</XMTok>
</XMApp>
<XMApp>
<XMTok mathstyle="display" meaning="integral" name="int" role="INTOP" xml:id="S8.Ex62.m1.35">∫</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.36">x</XMTok>
<XMTok font="italic" role="UNKNOWN" rpadding="1.7pt" xml:id="S8.Ex62.m1.37">y</XMTok>
<XMText xml:id="S8.Ex62.m1.38">dx</XMText>
</XMApp>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.40">x</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.41">y</XMTok>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.43">x</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.44">y</XMTok>
</XMApp>
</XMApp>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="right">
<XMApp>
<XMTok meaning="plus" role="ADDOP" xml:id="S8.Ex62.m1.48">+</XMTok>
<XMApp>
<XMTok meaning="plus" role="ADDOP" xml:id="S8.Ex62.m1.45">+</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.46">x</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.47">y</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.49">x</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.50">y</XMTok>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.52">x</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.53">y</XMTok>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.55">x</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S8.Ex62.m1.56">y</XMTok>
</XMApp>
</XMApp>
</XMCell>
</XMRow>
</XMArray>
</XMDual>
<XMTok font="italic" role="UNKNOWN">z</XMTok>
</XMApp>
</XMApp>
</XMath>
</Math>
</equation>
</para>
</section>
<section inlist="toc" xml:id="S9">
<tags>
<tag>9</tag>
<tag role="refnum">9</tag>
<tag role="typerefnum">§9</tag>
</tags>
<title><tag close=" ">9</tag>Multlines</title>
<para xml:id="S9.p1">
<equation>
<Math mode="display" tex="p(x)=3x^{6}+14x^{5}y+590x^{4}y^{2}+19x^{3}y^{3}\\ -12x^{2}y^{4}-12xy^{5}+2y^{6}-a^{3}b^{3}" text="p * x = (((3 * x ^ 6 + 14 * x ^ 5 * y + 590 * x ^ 4 * y ^ 2 + 19 * x ^ 3 * y ^ 3) - 12 * x ^ 2 * y ^ 4 - 12 * x * y ^ 5) + 2 * y ^ 6) - a ^ 3 * b ^ 3" xml:id="S9.p1.m1">
<XMath>
<XMDual>
<XMApp>
<XMRef idref="S9.p1.m1.5"/>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S9.p1.m1.1"/>
<XMRef idref="S9.p1.m1.3"/>
</XMApp>
<XMApp>
<XMRef idref="S9.p1.m1.41"/>
<XMApp>
<XMRef idref="S9.p1.m1.37"/>
<XMApp>
<XMRef idref="S9.p1.m1.26"/>
<XMApp>
<XMRef idref="S9.p1.m1.9"/>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S9.p1.m1.6"/>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMRef idref="S9.p1.m1.7"/>
<XMRef idref="S9.p1.m1.8.1"/>
</XMApp>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S9.p1.m1.10"/>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMRef idref="S9.p1.m1.11"/>
<XMRef idref="S9.p1.m1.12.1"/>
</XMApp>
<XMRef idref="S9.p1.m1.13"/>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S9.p1.m1.15"/>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMRef idref="S9.p1.m1.16"/>
<XMRef idref="S9.p1.m1.17.1"/>
</XMApp>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMRef idref="S9.p1.m1.18"/>
<XMRef idref="S9.p1.m1.19.1"/>
</XMApp>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S9.p1.m1.21"/>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMRef idref="S9.p1.m1.22"/>
<XMRef idref="S9.p1.m1.23.1"/>
</XMApp>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMRef idref="S9.p1.m1.24"/>
<XMRef idref="S9.p1.m1.25.1"/>
</XMApp>
</XMApp>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S9.p1.m1.27"/>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMRef idref="S9.p1.m1.28"/>
<XMRef idref="S9.p1.m1.29.1"/>
</XMApp>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMRef idref="S9.p1.m1.30"/>
<XMRef idref="S9.p1.m1.31.1"/>
</XMApp>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S9.p1.m1.33"/>
<XMRef idref="S9.p1.m1.34"/>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMRef idref="S9.p1.m1.35"/>
<XMRef idref="S9.p1.m1.36.1"/>
</XMApp>
</XMApp>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S9.p1.m1.38"/>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMRef idref="S9.p1.m1.39"/>
<XMRef idref="S9.p1.m1.40.1"/>
</XMApp>
</XMApp>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMRef idref="S9.p1.m1.42"/>
<XMRef idref="S9.p1.m1.43.1"/>
</XMApp>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMRef idref="S9.p1.m1.44"/>
<XMRef idref="S9.p1.m1.45.1"/>
</XMApp>
</XMApp>
</XMApp>
</XMApp>
<XMArray name="multline">
<XMRow>
<XMCell align="left">
<XMApp>
<XMTok meaning="equals" role="RELOP" xml:id="S9.p1.m1.5">=</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S9.p1.m1.1">p</XMTok>
<XMWrap>
<XMTok role="OPEN" stretchy="false" xml:id="S9.p1.m1.2">(</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S9.p1.m1.3">x</XMTok>
<XMTok role="CLOSE" stretchy="false" xml:id="S9.p1.m1.4">)</XMTok>
</XMWrap>
</XMApp>
<XMApp>
<XMTok meaning="plus" role="ADDOP" xml:id="S9.p1.m1.9">+</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok meaning="3" role="NUMBER" xml:id="S9.p1.m1.6">3</XMTok>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMTok font="italic" role="UNKNOWN" xml:id="S9.p1.m1.7">x</XMTok>
<XMTok fontsize="70%" meaning="6" role="NUMBER" xml:id="S9.p1.m1.8.1">6</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok meaning="14" role="NUMBER" xml:id="S9.p1.m1.10">14</XMTok>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
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<XMApp>
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<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
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<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMTok font="italic" role="UNKNOWN" xml:id="S9.p1.m1.18">y</XMTok>
<XMTok fontsize="70%" meaning="2" role="NUMBER" xml:id="S9.p1.m1.19.1">2</XMTok>
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<XMApp>
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<XMTok meaning="19" role="NUMBER" xml:id="S9.p1.m1.21">19</XMTok>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMTok font="italic" role="UNKNOWN" xml:id="S9.p1.m1.22">x</XMTok>
<XMTok fontsize="70%" meaning="3" role="NUMBER" xml:id="S9.p1.m1.23.1">3</XMTok>
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<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
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<XMTok fontsize="70%" meaning="3" role="NUMBER" xml:id="S9.p1.m1.25.1">3</XMTok>
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</XMApp>
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<XMRow>
<XMCell align="right">
<XMApp>
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<XMApp>
<XMTok meaning="plus" role="ADDOP" xml:id="S9.p1.m1.37">+</XMTok>
<XMApp>
<XMTok meaning="minus" role="ADDOP" xml:id="S9.p1.m1.32">-</XMTok>
<XMApp>
<XMTok meaning="minus" role="ADDOP" xml:id="S9.p1.m1.26">-</XMTok>
<XMApp>
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<XMTok meaning="12" role="NUMBER" xml:id="S9.p1.m1.27">12</XMTok>
<XMApp>
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<XMApp>
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<XMTok fontsize="70%" meaning="4" role="NUMBER" xml:id="S9.p1.m1.31.1">4</XMTok>
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</XMApp>
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<XMApp>
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<XMTok meaning="12" role="NUMBER" xml:id="S9.p1.m1.33">12</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S9.p1.m1.34">x</XMTok>
<XMApp>
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<XMTok font="italic" role="UNKNOWN" xml:id="S9.p1.m1.35">y</XMTok>
<XMTok fontsize="70%" meaning="5" role="NUMBER" xml:id="S9.p1.m1.36.1">5</XMTok>
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<XMApp>
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<XMTok meaning="2" role="NUMBER" xml:id="S9.p1.m1.38">2</XMTok>
<XMApp>
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<XMApp>
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<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMTok font="italic" role="UNKNOWN" xml:id="S9.p1.m1.42">a</XMTok>
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<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post5"/>
<XMTok font="italic" role="UNKNOWN" xml:id="S9.p1.m1.44">b</XMTok>
<XMTok fontsize="70%" meaning="3" role="NUMBER" xml:id="S9.p1.m1.45.1">3</XMTok>
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<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">A</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMDual>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S9.Ex63.m1.1"/>
<XMRef idref="S9.Ex63.m1.2"/>
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<XMArray name="multlined" vattach="top">
<XMRow>
<XMCell align="left">
<XMApp enclose="box" xml:id="S9.Ex63.m1.1">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">f</XMTok>
<XMTok font="italic" role="UNKNOWN">i</XMTok>
<XMTok font="italic" role="UNKNOWN">r</XMTok>
<XMTok font="italic" role="UNKNOWN">s</XMTok>
<XMTok font="italic" role="UNKNOWN">t</XMTok>
</XMApp>
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<XMRow>
<XMCell align="right">
<XMApp enclose="box" xml:id="S9.Ex63.m1.2">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">l</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">s</XMTok>
<XMTok font="italic" role="UNKNOWN">t</XMTok>
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<XMTok font="italic" role="UNKNOWN">B</XMTok>
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<equation xml:id="S9.Ex64">
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<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">A</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMDual>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S9.Ex64.m1.1"/>
<XMRef idref="S9.Ex64.m1.2"/>
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<XMArray name="multlined">
<XMRow>
<XMCell align="left">
<XMApp enclose="box" xml:id="S9.Ex64.m1.1">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">f</XMTok>
<XMTok font="italic" role="UNKNOWN">i</XMTok>
<XMTok font="italic" role="UNKNOWN">r</XMTok>
<XMTok font="italic" role="UNKNOWN">s</XMTok>
<XMTok font="italic" role="UNKNOWN">t</XMTok>
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<XMRow>
<XMCell align="right">
<XMApp enclose="box" xml:id="S9.Ex64.m1.2">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">l</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">s</XMTok>
<XMTok font="italic" role="UNKNOWN">t</XMTok>
</XMApp>
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</XMArray>
</XMDual>
<XMTok font="italic" role="UNKNOWN">B</XMTok>
</XMApp>
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<equation xml:id="S9.Ex65">
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<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">A</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMDual>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S9.Ex65.m1.1"/>
<XMRef idref="S9.Ex65.m1.2"/>
</XMApp>
<XMArray name="multlined" vattach="bottom">
<XMRow>
<XMCell align="left">
<XMApp enclose="box" xml:id="S9.Ex65.m1.1">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">f</XMTok>
<XMTok font="italic" role="UNKNOWN">i</XMTok>
<XMTok font="italic" role="UNKNOWN">r</XMTok>
<XMTok font="italic" role="UNKNOWN">s</XMTok>
<XMTok font="italic" role="UNKNOWN">t</XMTok>
</XMApp>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="right">
<XMApp enclose="box" xml:id="S9.Ex65.m1.2">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">l</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">s</XMTok>
<XMTok font="italic" role="UNKNOWN">t</XMTok>
</XMApp>
</XMCell>
</XMRow>
</XMArray>
</XMDual>
<XMTok font="italic" role="UNKNOWN">B</XMTok>
</XMApp>
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<equation xml:id="S9.Ex66">
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<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">A</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMDual>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S9.Ex66.m1.1"/>
<XMRef idref="S9.Ex66.m1.2"/>
</XMApp>
<XMArray name="multlined" vattach="middle">
<XMRow>
<XMCell align="left">
<XMApp enclose="box" xml:id="S9.Ex66.m1.1">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">f</XMTok>
<XMTok font="italic" role="UNKNOWN">i</XMTok>
<XMTok font="italic" role="UNKNOWN">r</XMTok>
<XMTok font="italic" role="UNKNOWN">s</XMTok>
<XMTok font="italic" role="UNKNOWN">t</XMTok>
</XMApp>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="right">
<XMApp enclose="box" xml:id="S9.Ex66.m1.2">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">l</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">s</XMTok>
<XMTok font="italic" role="UNKNOWN">t</XMTok>
</XMApp>
</XMCell>
</XMRow>
</XMArray>
</XMDual>
<XMTok font="italic" role="UNKNOWN">B</XMTok>
</XMApp>
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<equation xml:id="S9.Ex67">
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<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">A</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMDual>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S9.Ex67.m1.1"/>
<XMRef idref="S9.Ex67.m1.2"/>
</XMApp>
<XMArray name="multlined" vattach="middle">
<XMRow>
<XMCell align="left">
<XMApp enclose="box" xml:id="S9.Ex67.m1.1">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">f</XMTok>
<XMTok font="italic" role="UNKNOWN">i</XMTok>
<XMTok font="italic" role="UNKNOWN">r</XMTok>
<XMTok font="italic" role="UNKNOWN">s</XMTok>
<XMTok font="italic" role="UNKNOWN">t</XMTok>
</XMApp>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="right">
<XMApp enclose="box" xml:id="S9.Ex67.m1.2">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">l</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">s</XMTok>
<XMTok font="italic" role="UNKNOWN">t</XMTok>
</XMApp>
</XMCell>
</XMRow>
</XMArray>
</XMDual>
<XMTok font="italic" role="UNKNOWN">B</XMTok>
</XMApp>
</XMApp>
</XMath>
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<equation xml:id="S9.Ex68">
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<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">A</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMDual>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S9.Ex68.m1.1"/>
<XMRef idref="S9.Ex68.m1.2"/>
</XMApp>
<XMArray name="multlined" vattach="middle">
<XMRow>
<XMCell align="left">
<XMApp enclose="box" xml:id="S9.Ex68.m1.1">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">f</XMTok>
<XMTok font="italic" role="UNKNOWN">i</XMTok>
<XMTok font="italic" role="UNKNOWN">r</XMTok>
<XMTok font="italic" role="UNKNOWN">s</XMTok>
<XMTok font="italic" role="UNKNOWN">t</XMTok>
</XMApp>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="right">
<XMApp enclose="box" xml:id="S9.Ex68.m1.2">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">l</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">s</XMTok>
<XMTok font="italic" role="UNKNOWN">t</XMTok>
</XMApp>
</XMCell>
</XMRow>
</XMArray>
</XMDual>
<XMTok font="italic" role="UNKNOWN">B</XMTok>
</XMApp>
</XMApp>
</XMath>
</Math>
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<equation xml:id="S9.Ex69">
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<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">A</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMDual>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S9.Ex69.m1.1"/>
<XMRef idref="S9.Ex69.m1.2"/>
</XMApp>
<XMArray name="multlined" vattach="top">
<XMRow>
<XMCell align="left">
<XMApp enclose="box" xml:id="S9.Ex69.m1.1">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">f</XMTok>
<XMTok font="italic" role="UNKNOWN">i</XMTok>
<XMTok font="italic" role="UNKNOWN">r</XMTok>
<XMTok font="italic" role="UNKNOWN">s</XMTok>
<XMTok font="italic" role="UNKNOWN">t</XMTok>
</XMApp>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="right">
<XMApp enclose="box" xml:id="S9.Ex69.m1.2">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">l</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">s</XMTok>
<XMTok font="italic" role="UNKNOWN">t</XMTok>
</XMApp>
</XMCell>
</XMRow>
</XMArray>
</XMDual>
<XMTok font="italic" role="UNKNOWN">B</XMTok>
</XMApp>
</XMApp>
</XMath>
</Math>
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<equation xml:id="S9.Ex70">
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<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">A</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMDual>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S9.Ex70.m1.1"/>
<XMRef idref="S9.Ex70.m1.2"/>
</XMApp>
<XMArray name="multlined" vattach="bottom">
<XMRow>
<XMCell align="left">
<XMApp enclose="box" xml:id="S9.Ex70.m1.1">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">f</XMTok>
<XMTok font="italic" role="UNKNOWN">i</XMTok>
<XMTok font="italic" role="UNKNOWN">r</XMTok>
<XMTok font="italic" role="UNKNOWN">s</XMTok>
<XMTok font="italic" role="UNKNOWN">t</XMTok>
</XMApp>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="right">
<XMApp enclose="box" xml:id="S9.Ex70.m1.2">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">l</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">s</XMTok>
<XMTok font="italic" role="UNKNOWN">t</XMTok>
</XMApp>
</XMCell>
</XMRow>
</XMArray>
</XMDual>
<XMTok font="italic" role="UNKNOWN">B</XMTok>
</XMApp>
</XMApp>
</XMath>
</Math>
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<equation xml:id="S9.Ex71">
<Math mode="display" tex="A=\begin{multlined}\boxed{first}\\ \boxed{last}\end{multlined}\boxed{first}\\ \boxed{last}B" text="A = f * i * r * s * t * l * a * s * t * B" xml:id="S9.Ex71.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">A</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMDual>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S9.Ex71.m1.1"/>
<XMRef idref="S9.Ex71.m1.2"/>
</XMApp>
<XMArray name="multlined" vattach="middle">
<XMRow>
<XMCell align="left">
<XMApp enclose="box" xml:id="S9.Ex71.m1.1">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">f</XMTok>
<XMTok font="italic" role="UNKNOWN">i</XMTok>
<XMTok font="italic" role="UNKNOWN">r</XMTok>
<XMTok font="italic" role="UNKNOWN">s</XMTok>
<XMTok font="italic" role="UNKNOWN">t</XMTok>
</XMApp>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="right">
<XMApp enclose="box" xml:id="S9.Ex71.m1.2">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">l</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">s</XMTok>
<XMTok font="italic" role="UNKNOWN">t</XMTok>
</XMApp>
</XMCell>
</XMRow>
</XMArray>
</XMDual>
<XMTok font="italic" role="UNKNOWN">B</XMTok>
</XMApp>
</XMApp>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S9.p11">
<equation xml:id="S9.E14">
<tags>
<tag>(14)</tag>
<tag role="refnum">14</tag>
</tags>
<Math class="ltx_math_unparsed" mode="display" tex="foo\Coloneqq\begin{lgathered}x=1,\quad x+1=2\\ y=2\end{lgathered}x=1,\quad x+1=2\\ y=2" text="f * o * o Coloneqq x@=@1@,@quad@x@+@1@=@2@y@=@2" xml:id="S9.E14.m1">
<XMath>
<XMApp>
<XMTok name="Coloneqq" role="RELOP">⩴</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">f</XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
</XMApp>
<XMDual>
<XMWrap rule="Anything,">
<XMRef idref="S9.E14.m1.1"/>
<XMRef idref="S9.E14.m1.2"/>
<XMRef idref="S9.E14.m1.3"/>
<XMRef idref="S9.E14.m1.4"/>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMRef idref="S9.E14.m1.5"/>
<XMRef idref="S9.E14.m1.6"/>
<XMRef idref="S9.E14.m1.7"/>
<XMRef idref="S9.E14.m1.8"/>
<XMRef idref="S9.E14.m1.9"/>
<XMRef idref="S9.E14.m1.10"/>
<XMRef idref="S9.E14.m1.11"/>
<XMRef idref="S9.E14.m1.12"/>
</XMWrap>
<XMArray name="lgathered" vattach="top">
<XMRow>
<XMCell align="left">
<XMWrap>
<XMApp xml:id="S9.E14.m1.13">
<XMTok meaning="equals" role="RELOP" xml:id="S9.E14.m1.2">=</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S9.E14.m1.1">x</XMTok>
<XMTok meaning="1" role="NUMBER" xml:id="S9.E14.m1.3">1</XMTok>
</XMApp>
<XMTok role="PUNCT" rpadding="10.0pt" xml:id="S9.E14.m1.4">,</XMTok>
<XMApp xml:id="S9.E14.m1.14">
<XMTok meaning="equals" role="RELOP" xml:id="S9.E14.m1.8">=</XMTok>
<XMApp>
<XMTok meaning="plus" role="ADDOP" xml:id="S9.E14.m1.6">+</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S9.E14.m1.5">x</XMTok>
<XMTok meaning="1" role="NUMBER" xml:id="S9.E14.m1.7">1</XMTok>
</XMApp>
<XMTok meaning="2" role="NUMBER" xml:id="S9.E14.m1.9">2</XMTok>
</XMApp>
</XMWrap>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="left">
<XMApp>
<XMTok meaning="equals" role="RELOP" xml:id="S9.E14.m1.11">=</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S9.E14.m1.10">y</XMTok>
<XMTok meaning="2" role="NUMBER" xml:id="S9.E14.m1.12">2</XMTok>
</XMApp>
</XMCell>
</XMRow>
</XMArray>
</XMDual>
</XMApp>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S9.p12">
<equation xml:id="S9.E15">
<tags>
<tag>(15)</tag>
<tag role="refnum">15</tag>
</tags>
<Math class="ltx_math_unparsed" mode="display" tex="bar\Coloneqq\begin{rgathered}x=1,\quad x+1=2\\ y=2\end{rgathered}x=1,\quad x+1=2\\ y=2" text="b * a * r Coloneqq x@=@1@,@quad@x@+@1@=@2@y@=@2" xml:id="S9.E15.m1">
<XMath>
<XMApp>
<XMTok name="Coloneqq" role="RELOP">⩴</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">r</XMTok>
</XMApp>
<XMDual>
<XMWrap rule="Anything,">
<XMRef idref="S9.E15.m1.1"/>
<XMRef idref="S9.E15.m1.2"/>
<XMRef idref="S9.E15.m1.3"/>
<XMRef idref="S9.E15.m1.4"/>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMRef idref="S9.E15.m1.5"/>
<XMRef idref="S9.E15.m1.6"/>
<XMRef idref="S9.E15.m1.7"/>
<XMRef idref="S9.E15.m1.8"/>
<XMRef idref="S9.E15.m1.9"/>
<XMRef idref="S9.E15.m1.10"/>
<XMRef idref="S9.E15.m1.11"/>
<XMRef idref="S9.E15.m1.12"/>
</XMWrap>
<XMArray name="rgathered" vattach="bottom">
<XMRow>
<XMCell align="right">
<XMWrap>
<XMApp xml:id="S9.E15.m1.13">
<XMTok meaning="equals" role="RELOP" xml:id="S9.E15.m1.2">=</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S9.E15.m1.1">x</XMTok>
<XMTok meaning="1" role="NUMBER" xml:id="S9.E15.m1.3">1</XMTok>
</XMApp>
<XMTok role="PUNCT" rpadding="10.0pt" xml:id="S9.E15.m1.4">,</XMTok>
<XMApp xml:id="S9.E15.m1.14">
<XMTok meaning="equals" role="RELOP" xml:id="S9.E15.m1.8">=</XMTok>
<XMApp>
<XMTok meaning="plus" role="ADDOP" xml:id="S9.E15.m1.6">+</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S9.E15.m1.5">x</XMTok>
<XMTok meaning="1" role="NUMBER" xml:id="S9.E15.m1.7">1</XMTok>
</XMApp>
<XMTok meaning="2" role="NUMBER" xml:id="S9.E15.m1.9">2</XMTok>
</XMApp>
</XMWrap>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="right">
<XMApp>
<XMTok meaning="equals" role="RELOP" xml:id="S9.E15.m1.11">=</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S9.E15.m1.10">y</XMTok>
<XMTok meaning="2" role="NUMBER" xml:id="S9.E15.m1.12">2</XMTok>
</XMApp>
</XMCell>
</XMRow>
</XMArray>
</XMDual>
</XMApp>
</XMath>
</Math>
</equation>
</para>
</section>
<section inlist="toc" xml:id="S10">
<tags>
<tag>10</tag>
<tag role="refnum">10</tag>
<tag role="typerefnum">§10</tag>
</tags>
<title><tag close=" ">10</tag>Spread-lines</title>
<para xml:id="S10.p1">
<p>Spread it</p>
<equation xml:id="S10.Ex72">
<Math mode="display" tex="\begin{matrix}a&b&c\\ d&e&f\\ g&h&i\end{matrix}" text="matrix@(Array[[a, b, c], [d, e, f], [g, h, i]])" xml:id="S10.Ex72.m1">
<XMath>
<XMDual>
<XMApp>
<XMTok meaning="matrix"/>
<XMRef idref="S10.Ex72.m1.1"/>
</XMApp>
<XMArray xml:id="S10.Ex72.m1.1">
<XMRow>
<XMCell align="center">
<XMTok font="italic" role="UNKNOWN">a</XMTok>
</XMCell>
<XMCell align="center">
<XMTok font="italic" role="UNKNOWN">b</XMTok>
</XMCell>
<XMCell align="center">
<XMTok font="italic" role="UNKNOWN">c</XMTok>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="center">
<XMTok font="italic" role="UNKNOWN">d</XMTok>
</XMCell>
<XMCell align="center">
<XMTok font="italic" role="UNKNOWN">e</XMTok>
</XMCell>
<XMCell align="center">
<XMTok font="italic" role="UNKNOWN">f</XMTok>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="center">
<XMTok font="italic" role="UNKNOWN">g</XMTok>
</XMCell>
<XMCell align="center">
<XMTok font="italic" role="UNKNOWN">h</XMTok>
</XMCell>
<XMCell align="center">
<XMTok font="italic" role="UNKNOWN">i</XMTok>
</XMCell>
</XMRow>
</XMArray>
</XMDual>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S10.p2">
<equation xml:id="S10.Ex73">
<Math mode="display" tex="\begin{pmatrix}a_{1,1}&a_{1,2}&\cdots&a_{1,n}\\ a_{2,1}&a_{2,2}&\cdots&a_{2,n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{m,1}&a_{m,2}&\cdots&a_{m,n}\end{pmatrix}" text="matrix@(Array[[a _ (list@(1, 1)), a _ (list@(1, 2)), cdots, a _ (list@(1, n))], [a _ (list@(2, 1)), a _ (list@(2, 2)), cdots, a _ (list@(2, n))], [vdots, vdots, ddots, vdots], [a _ (list@(m, 1)), a _ (list@(m, 2)), cdots, a _ (list@(m, n))]])" xml:id="S10.Ex73.m1">
<XMath>
<XMDual>
<XMApp>
<XMTok meaning="matrix"/>
<XMRef idref="S10.Ex73.m1.1"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="true">(</XMTok>
<XMArray xml:id="S10.Ex73.m1.1">
<XMRow>
<XMCell align="center">
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post5"/>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMDual>
<XMApp>
<XMTok meaning="list"/>
<XMRef idref="S10.Ex73.m1.1.1"/>
<XMRef idref="S10.Ex73.m1.1.2"/>
</XMApp>
<XMWrap>
<XMTok fontsize="70%" meaning="1" role="NUMBER" xml:id="S10.Ex73.m1.1.1">1</XMTok>
<XMTok fontsize="70%" role="PUNCT">,</XMTok>
<XMTok fontsize="70%" meaning="1" role="NUMBER" xml:id="S10.Ex73.m1.1.2">1</XMTok>
</XMWrap>
</XMDual>
</XMApp>
</XMCell>
<XMCell align="center">
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post5"/>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMDual>
<XMApp>
<XMTok meaning="list"/>
<XMRef idref="S10.Ex73.m1.1.3"/>
<XMRef idref="S10.Ex73.m1.1.4"/>
</XMApp>
<XMWrap>
<XMTok fontsize="70%" meaning="1" role="NUMBER" xml:id="S10.Ex73.m1.1.3">1</XMTok>
<XMTok fontsize="70%" role="PUNCT">,</XMTok>
<XMTok fontsize="70%" meaning="2" role="NUMBER" xml:id="S10.Ex73.m1.1.4">2</XMTok>
</XMWrap>
</XMDual>
</XMApp>
</XMCell>
<XMCell align="center">
<XMTok name="cdots" role="ID">⋯</XMTok>
</XMCell>
<XMCell align="center">
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post5"/>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMDual>
<XMApp>
<XMTok meaning="list"/>
<XMRef idref="S10.Ex73.m1.1.5"/>
<XMRef idref="S10.Ex73.m1.1.6"/>
</XMApp>
<XMWrap>
<XMTok fontsize="70%" meaning="1" role="NUMBER" xml:id="S10.Ex73.m1.1.5">1</XMTok>
<XMTok fontsize="70%" role="PUNCT">,</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN" xml:id="S10.Ex73.m1.1.6">n</XMTok>
</XMWrap>
</XMDual>
</XMApp>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="center">
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post5"/>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMDual>
<XMApp>
<XMTok meaning="list"/>
<XMRef idref="S10.Ex73.m1.1.7"/>
<XMRef idref="S10.Ex73.m1.1.8"/>
</XMApp>
<XMWrap>
<XMTok fontsize="70%" meaning="2" role="NUMBER" xml:id="S10.Ex73.m1.1.7">2</XMTok>
<XMTok fontsize="70%" role="PUNCT">,</XMTok>
<XMTok fontsize="70%" meaning="1" role="NUMBER" xml:id="S10.Ex73.m1.1.8">1</XMTok>
</XMWrap>
</XMDual>
</XMApp>
</XMCell>
<XMCell align="center">
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post5"/>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMDual>
<XMApp>
<XMTok meaning="list"/>
<XMRef idref="S10.Ex73.m1.1.9"/>
<XMRef idref="S10.Ex73.m1.1.10"/>
</XMApp>
<XMWrap>
<XMTok fontsize="70%" meaning="2" role="NUMBER" xml:id="S10.Ex73.m1.1.9">2</XMTok>
<XMTok fontsize="70%" role="PUNCT">,</XMTok>
<XMTok fontsize="70%" meaning="2" role="NUMBER" xml:id="S10.Ex73.m1.1.10">2</XMTok>
</XMWrap>
</XMDual>
</XMApp>
</XMCell>
<XMCell align="center">
<XMTok name="cdots" role="ID">⋯</XMTok>
</XMCell>
<XMCell align="center">
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post5"/>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMDual>
<XMApp>
<XMTok meaning="list"/>
<XMRef idref="S10.Ex73.m1.1.11"/>
<XMRef idref="S10.Ex73.m1.1.12"/>
</XMApp>
<XMWrap>
<XMTok fontsize="70%" meaning="2" role="NUMBER" xml:id="S10.Ex73.m1.1.11">2</XMTok>
<XMTok fontsize="70%" role="PUNCT">,</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN" xml:id="S10.Ex73.m1.1.12">n</XMTok>
</XMWrap>
</XMDual>
</XMApp>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="center">
<XMTok name="vdots" role="ID">⋮</XMTok>
</XMCell>
<XMCell align="center">
<XMTok name="vdots" role="ID">⋮</XMTok>
</XMCell>
<XMCell align="center">
<XMTok name="ddots" role="ID">⋱</XMTok>
</XMCell>
<XMCell align="center">
<XMTok name="vdots" role="ID">⋮</XMTok>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="center">
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post5"/>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMDual>
<XMApp>
<XMTok meaning="list"/>
<XMRef idref="S10.Ex73.m1.1.13"/>
<XMRef idref="S10.Ex73.m1.1.14"/>
</XMApp>
<XMWrap>
<XMTok font="italic" fontsize="70%" role="UNKNOWN" xml:id="S10.Ex73.m1.1.13">m</XMTok>
<XMTok fontsize="70%" role="PUNCT">,</XMTok>
<XMTok fontsize="70%" meaning="1" role="NUMBER" xml:id="S10.Ex73.m1.1.14">1</XMTok>
</XMWrap>
</XMDual>
</XMApp>
</XMCell>
<XMCell align="center">
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post5"/>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMDual>
<XMApp>
<XMTok meaning="list"/>
<XMRef idref="S10.Ex73.m1.1.15"/>
<XMRef idref="S10.Ex73.m1.1.16"/>
</XMApp>
<XMWrap>
<XMTok font="italic" fontsize="70%" role="UNKNOWN" xml:id="S10.Ex73.m1.1.15">m</XMTok>
<XMTok fontsize="70%" role="PUNCT">,</XMTok>
<XMTok fontsize="70%" meaning="2" role="NUMBER" xml:id="S10.Ex73.m1.1.16">2</XMTok>
</XMWrap>
</XMDual>
</XMApp>
</XMCell>
<XMCell align="center">
<XMTok name="cdots" role="ID">⋯</XMTok>
</XMCell>
<XMCell align="center">
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post5"/>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMDual>
<XMApp>
<XMTok meaning="list"/>
<XMRef idref="S10.Ex73.m1.1.17"/>
<XMRef idref="S10.Ex73.m1.1.18"/>
</XMApp>
<XMWrap>
<XMTok font="italic" fontsize="70%" role="UNKNOWN" xml:id="S10.Ex73.m1.1.17">m</XMTok>
<XMTok fontsize="70%" role="PUNCT">,</XMTok>
<XMTok font="italic" fontsize="70%" role="UNKNOWN" xml:id="S10.Ex73.m1.1.18">n</XMTok>
</XMWrap>
</XMDual>
</XMApp>
</XMCell>
</XMRow>
</XMArray>
<XMTok role="CLOSE" stretchy="true">)</XMTok>
</XMWrap>
</XMDual>
</XMath>
</Math>
</equation>
</para>
<para xml:id="S10.p3">
<p><Math mode="inline" tex="\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}" text="Array[[a, b], [c, d]]" xml:id="S10.p3.m1">
<XMath>
<XMArray>
<XMRow>
<XMCell align="center">
<XMTok font="italic" fontsize="70%" role="UNKNOWN">a</XMTok>
</XMCell>
<XMCell align="center">
<XMTok font="italic" fontsize="70%" role="UNKNOWN">b</XMTok>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="center">
<XMTok font="italic" fontsize="70%" role="UNKNOWN">c</XMTok>
</XMCell>
<XMCell align="center">
<XMTok font="italic" fontsize="70%" role="UNKNOWN">d</XMTok>
</XMCell>
</XMRow>
</XMArray>
</XMath>
</Math></p>
</para>
<para xml:id="S10.p4">
<p><Math mode="inline" tex="\begin{cases}n/2&\quad\text{if }n\text{ is even}\\ -(n+1)/2&\quad\text{if }n\text{ is odd}\\ \end{cases}" text="cases@(n / 2, [if ] * n * [ is even], - (n + 1) / 2, [if ] * n * [ is odd])" xml:id="S10.p4.m1">
<XMath>
<XMDual>
<XMApp>
<XMTok meaning="cases"/>
<XMRef idref="S10.p4.m1.1"/>
<XMRef idref="S10.p4.m1.2"/>
<XMRef idref="S10.p4.m1.3"/>
<XMRef idref="S10.p4.m1.4"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="true">{</XMTok>
<XMArray>
<XMRow>
<XMCell align="left">
<XMApp xml:id="S10.p4.m1.1">
<XMTok meaning="divide" role="MULOP">/</XMTok>
<XMTok font="italic" role="UNKNOWN">n</XMTok>
<XMTok meaning="2" role="NUMBER">2</XMTok>
</XMApp>
</XMCell>
<XMCell align="left">
<XMApp xml:id="S10.p4.m1.2">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMText lpadding="10.0pt">if </XMText>
<XMTok font="italic" role="UNKNOWN">n</XMTok>
<XMText> is even</XMText>
</XMApp>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="left">
<XMApp xml:id="S10.p4.m1.3">
<XMTok meaning="minus" role="ADDOP">-</XMTok>
<XMApp>
<XMTok meaning="divide" role="MULOP">/</XMTok>
<XMDual>
<XMRef idref="S10.p4.m1.3.1"/>
<XMWrap>
<XMTok role="OPEN" stretchy="false">(</XMTok>
<XMApp xml:id="S10.p4.m1.3.1">
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMTok font="italic" role="UNKNOWN">n</XMTok>
<XMTok meaning="1" role="NUMBER">1</XMTok>
</XMApp>
<XMTok role="CLOSE" stretchy="false">)</XMTok>
</XMWrap>
</XMDual>
<XMTok meaning="2" role="NUMBER">2</XMTok>
</XMApp>
</XMApp>
</XMCell>
<XMCell align="left">
<XMApp xml:id="S10.p4.m1.4">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMText lpadding="10.0pt">if </XMText>
<XMTok font="italic" role="UNKNOWN">n</XMTok>
<XMText> is odd</XMText>
</XMApp>
</XMCell>
</XMRow>
</XMArray>
</XMWrap>
</XMDual>
</XMath>
</Math></p>
</para>
<para xml:id="S10.p5">
<equation labels="LABEL:xx" xml:id="S10.E16">
<tags>
<tag>(16)</tag>
<tag role="refnum">16</tag>
</tags>
<Math mode="display" tex="\begin{split}a&=b+c-d\\ &\quad+e-f\\ &=g+h\\ &=i\end{split}" text="a = (((b + c) - d) + e) - f = g + h = i" xml:id="S10.E16.m1">
<XMath>
<XMDual>
<XMApp>
<XMTok meaning="multirelation"/>
<XMRef idref="S10.E16.m1.1"/>
<XMRef idref="S10.E16.m1.2"/>
<XMApp>
<XMRef idref="S10.E16.m1.10"/>
<XMApp>
<XMRef idref="S10.E16.m1.8"/>
<XMApp>
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<XMRef idref="S10.E16.m1.3"/>
<XMRef idref="S10.E16.m1.5"/>
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<XMRef idref="S10.E16.m1.7"/>
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<XMRef idref="S10.E16.m1.9"/>
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<XMRef idref="S10.E16.m1.11"/>
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<XMRef idref="S10.E16.m1.12"/>
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<XMRef idref="S10.E16.m1.13"/>
<XMRef idref="S10.E16.m1.15"/>
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<XMRef idref="S10.E16.m1.17"/>
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<XMRow>
<XMCell align="right">
<XMTok font="italic" role="UNKNOWN" xml:id="S10.E16.m1.1">a</XMTok>
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<XMTok font="italic" role="UNKNOWN" xml:id="S10.E16.m1.5">c</XMTok>
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<XMTok font="italic" role="UNKNOWN" xml:id="S10.E16.m1.7">d</XMTok>
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</XMApp>
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<XMRow>
<XMCell/>
<XMCell align="left">
<XMApp>
<XMTok meaning="minus" role="ADDOP" xml:id="S10.E16.m1.10">-</XMTok>
<XMApp>
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<XMTok font="italic" role="UNKNOWN" xml:id="S10.E16.m1.11">f</XMTok>
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<XMRow>
<XMCell/>
<XMCell align="left">
<XMApp>
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<XMTok meaning="absent"/>
<XMApp>
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<XMTok font="italic" role="UNKNOWN" xml:id="S10.E16.m1.13">g</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S10.E16.m1.15">h</XMTok>
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</XMApp>
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<XMRow>
<XMCell/>
<XMCell align="left">
<XMApp>
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<equation xml:id="S10.E17">
<tags>
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<tag role="refnum">17</tag>
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<Math mode="display" tex="a+b+c+d+e+f\\ +i+j+k+l+m+n" text="a + b + c + d + e + f + i + j + k + l + m + n" xml:id="S10.E17.m1">
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<XMApp>
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<XMRef idref="S10.E17.m1.1"/>
<XMRef idref="S10.E17.m1.3"/>
<XMRef idref="S10.E17.m1.5"/>
<XMRef idref="S10.E17.m1.7"/>
<XMRef idref="S10.E17.m1.9"/>
<XMRef idref="S10.E17.m1.11"/>
<XMRef idref="S10.E17.m1.13"/>
<XMRef idref="S10.E17.m1.15"/>
<XMRef idref="S10.E17.m1.17"/>
<XMRef idref="S10.E17.m1.19"/>
<XMRef idref="S10.E17.m1.21"/>
<XMRef idref="S10.E17.m1.23"/>
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<XMArray name="multline" rowsep="30.0pt">
<XMRow>
<XMCell align="left">
<XMApp>
<XMTok meaning="plus" role="ADDOP" xml:id="S10.E17.m1.2">+</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S10.E17.m1.1">a</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S10.E17.m1.3">b</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S10.E17.m1.5">c</XMTok>
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<XMTok font="italic" role="UNKNOWN" xml:id="S10.E17.m1.11">f</XMTok>
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<XMApp>
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<XMTok font="italic" role="UNKNOWN" xml:id="S10.E17.m1.13">i</XMTok>
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<XMTok font="italic" role="UNKNOWN" xml:id="S10.E17.m1.15">j</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S10.E17.m1.17">k</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S10.E17.m1.19">l</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S10.E17.m1.21">m</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S10.E17.m1.23">n</XMTok>
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<para xml:id="S10.p7">
<equationgroup class="ltx_eqn_gather" rowsep="30.0pt" xml:id="S12.EGx5">
<equation xml:id="S10.E18">
<tags>
<tag>(18)</tag>
<tag role="refnum">18</tag>
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<Math mode="display" tex="\displaystyle a=b" text="a = b" xml:id="S10.E18.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
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</XMath>
</Math>
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<equation xml:id="S10.E19">
<tags>
<tag>(19)</tag>
<tag role="refnum">19</tag>
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<Math mode="display" tex="\displaystyle c=d" text="c = d" xml:id="S10.E19.m1">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
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</XMath>
</Math>
</equation>
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</para>
<para xml:id="S10.p8">
<equationgroup class="ltx_eqn_align" rowsep="30.0pt" xml:id="S12.EGx6">
<equation xml:id="S10.E20">
<tags>
<tag>(20)</tag>
<tag role="refnum">20</tag>
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<MathFork>
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<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMApp>
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<XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
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<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
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<XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
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<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
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</XMApp>
</XMApp>
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<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
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</XMath>
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<td align="left"><Math mode="inline" tex="\displaystyle=b_{1}+c_{1}" text="absent = b _ 1 + c _ 1" xml:id="S10.E20.m2">
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<XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
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<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
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</XMApp>
</XMApp>
</XMath>
</Math></td>
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<equation xml:id="S10.E21">
<tags>
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<tag role="refnum">21</tag>
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<MathFork>
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<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
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<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMApp>
<XMTok meaning="minus" role="ADDOP">-</XMTok>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
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</XMApp>
</XMApp>
</XMath>
</Math>
<MathBranch>
<td align="right"><Math mode="inline" tex="\displaystyle a_{2}" text="a _ 2" xml:id="S10.E21.m1">
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<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
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</XMath>
</Math></td>
<td align="left"><Math mode="inline" tex="\displaystyle=b_{2}+c_{2}-d_{2}+e_{2}" text="absent = ((b _ 2 + c _ 2) - d _ 2) + e _ 2" xml:id="S10.E21.m2">
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<XMApp>
<XMTok meaning="minus" role="ADDOP">-</XMTok>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
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<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
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<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
</XMApp>
</XMApp>
<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
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<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
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</XMApp>
</XMApp>
</XMath>
</Math></td>
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</para>
<para xml:id="S10.p9">
<equationgroup class="ltx_eqn_align" rowsep="30.0pt" xml:id="S12.EGx7">
<equation xml:id="S10.Ex74">
<MathFork>
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<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
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<XMTok fontsize="70%" meaning="11" role="NUMBER">11</XMTok>
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<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
<XMTok fontsize="70%" meaning="11" role="NUMBER">11</XMTok>
</XMApp>
</XMApp>
</XMath>
</Math>
<MathBranch>
<td align="right"><Math mode="inline" tex="\displaystyle a_{11}" text="a _ 11" xml:id="S10.Ex74.m1">
<XMath>
<XMApp>
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<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok fontsize="70%" meaning="11" role="NUMBER">11</XMTok>
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</XMath>
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<td align="left"><Math mode="inline" tex="\displaystyle=b_{11}" text="absent = b _ 11" xml:id="S10.Ex74.m2">
<XMath>
<XMApp>
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<XMTok meaning="absent"/>
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<XMTok fontsize="70%" meaning="11" role="NUMBER">11</XMTok>
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<MathFork>
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<XMTok fontsize="70%" meaning="12" role="NUMBER">12</XMTok>
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<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
<XMTok fontsize="70%" meaning="12" role="NUMBER">12</XMTok>
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</XMApp>
</XMath>
</Math>
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<td align="right"><Math mode="inline" tex="\displaystyle a_{12}" text="a _ 12" xml:id="S10.Ex74.m3">
<XMath>
<XMApp>
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<XMTok fontsize="70%" meaning="12" role="NUMBER">12</XMTok>
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</XMath>
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<td align="left"><Math mode="inline" tex="\displaystyle=b_{12}" text="absent = b _ 12" xml:id="S10.Ex74.m4">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok meaning="absent"/>
<XMApp>
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<XMTok fontsize="70%" meaning="12" role="NUMBER">12</XMTok>
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<MathFork>
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<XMApp>
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<XMTok fontsize="70%" meaning="21" role="NUMBER">21</XMTok>
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<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
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</XMApp>
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</XMath>
</Math></td>
<td align="left"><Math mode="inline" tex="\displaystyle=b_{21}" text="absent = b _ 21" xml:id="S10.Ex75.m2">
<XMath>
<XMApp>
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<XMTok fontsize="70%" meaning="22" role="NUMBER">22</XMTok>
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</XMApp>
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</XMath>
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<td align="left"><Math mode="inline" tex="\displaystyle=b_{22}+c_{22}" text="absent = b _ 22 + c _ 22" xml:id="S10.Ex75.m4">
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<XMApp>
<XMTok role="SUBSCRIPTOP" scriptpos="post1"/>
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<XMTok fontsize="70%" meaning="22" role="NUMBER">22</XMTok>
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<XMApp>
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<XMTok fontsize="70%" meaning="22" role="NUMBER">22</XMTok>
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</XMApp>
</XMath>
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</equationgroup>
</para>
<para xml:id="S10.p10">
<equationgroup class="ltx_eqn_align" rowsep="30.0pt" xml:id="S12.EGx8">
<equation xml:id="S10.E22">
<tags>
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<tag role="refnum">22</tag>
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<XMApp>
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<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok meaning="2" role="NUMBER">2</XMTok>
<XMTok font="italic" role="UNKNOWN">x</XMTok>
<XMTok font="italic" role="UNKNOWN">y</XMTok>
</XMApp>
<XMApp>
<XMTok role="SUPERSCRIPTOP" scriptpos="post1"/>
<XMTok font="italic" role="UNKNOWN">y</XMTok>
<XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
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</XMApp>
</XMath>
</Math></td>
</tr>
</MathBranch>
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</para>
<para xml:id="S10.p16">
<equation xml:id="S10.Ex80">
<Math mode="display" tex="\begin{multlined}\boxed{Firstline}\\ \boxed{Secondline}\\ L+E+F+T\\ R+I+G+H+T\\ L+E+F+T\\ R+I+G+H+T\\ \boxed{WupWup}\\ \boxed{Lastline}\end{multlined}\boxed{Firstline}\\ \boxed{Secondline}\\ L+E+F+T\\ R+I+G+H+T\\ L+E+F+T\\ R+I+G+H+T\\ \boxed{WupWup}\\ \boxed{Lastline}" text="F * i * r * s * t * l * i * n * e * S * e * c * o * n * d * l * i * n * e * L + E + F + T * R + I + G + H + T * L + E + F + T * R + I + G + H + T * W * u * p * W * u * p * L * a * s * t * l * i * n * e" xml:id="S10.Ex80.m1">
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<XMDual>
<XMApp>
<XMRef idref="S10.Ex80.m1.4"/>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
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<XMRef idref="S10.Ex80.m1.2"/>
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<XMRef idref="S10.Ex80.m1.5"/>
<XMRef idref="S10.Ex80.m1.7"/>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
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<XMRef idref="S10.Ex80.m1.12"/>
<XMRef idref="S10.Ex80.m1.14"/>
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<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S10.Ex80.m1.18"/>
<XMRef idref="S10.Ex80.m1.19"/>
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<XMRef idref="S10.Ex80.m1.21"/>
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<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S10.Ex80.m1.25"/>
<XMRef idref="S10.Ex80.m1.26"/>
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<XMRef idref="S10.Ex80.m1.28"/>
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<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMRef idref="S10.Ex80.m1.34"/>
<XMRef idref="S10.Ex80.m1.35"/>
<XMRef idref="S10.Ex80.m1.36"/>
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<XMCell align="left">
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<XMTok font="italic" role="UNKNOWN">F</XMTok>
<XMTok font="italic" role="UNKNOWN">i</XMTok>
<XMTok font="italic" role="UNKNOWN">r</XMTok>
<XMTok font="italic" role="UNKNOWN">s</XMTok>
<XMTok font="italic" role="UNKNOWN">t</XMTok>
<XMTok font="italic" role="UNKNOWN">l</XMTok>
<XMTok font="italic" role="UNKNOWN">i</XMTok>
<XMTok font="italic" role="UNKNOWN">n</XMTok>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
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</XMCell>
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<XMRow>
<XMCell align="center">
<XMApp enclose="box" xml:id="S10.Ex80.m1.2">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">S</XMTok>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
<XMTok font="italic" role="UNKNOWN">o</XMTok>
<XMTok font="italic" role="UNKNOWN">n</XMTok>
<XMTok font="italic" role="UNKNOWN">d</XMTok>
<XMTok font="italic" role="UNKNOWN">l</XMTok>
<XMTok font="italic" role="UNKNOWN">i</XMTok>
<XMTok font="italic" role="UNKNOWN">n</XMTok>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
</XMApp>
</XMCell>
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<XMRow>
<XMCell align="left">
<XMApp>
<XMTok meaning="plus" role="ADDOP" xml:id="S10.Ex80.m1.4">+</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S10.Ex80.m1.3">L</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S10.Ex80.m1.5">E</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S10.Ex80.m1.7">F</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S10.Ex80.m1.9">T</XMTok>
</XMApp>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="right">
<XMApp>
<XMTok meaning="plus" role="ADDOP" xml:id="S10.Ex80.m1.11">+</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S10.Ex80.m1.10">R</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S10.Ex80.m1.12">I</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S10.Ex80.m1.14">G</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S10.Ex80.m1.16">H</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S10.Ex80.m1.18">T</XMTok>
</XMApp>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="left">
<XMApp>
<XMTok meaning="plus" role="ADDOP" xml:id="S10.Ex80.m1.20">+</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S10.Ex80.m1.19">L</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S10.Ex80.m1.21">E</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S10.Ex80.m1.23">F</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S10.Ex80.m1.25">T</XMTok>
</XMApp>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="right">
<XMApp>
<XMTok meaning="plus" role="ADDOP" xml:id="S10.Ex80.m1.27">+</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S10.Ex80.m1.26">R</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S10.Ex80.m1.28">I</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S10.Ex80.m1.30">G</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S10.Ex80.m1.32">H</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S10.Ex80.m1.34">T</XMTok>
</XMApp>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="center">
<XMApp enclose="box" xml:id="S10.Ex80.m1.35">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">W</XMTok>
<XMTok font="italic" role="UNKNOWN">u</XMTok>
<XMTok font="italic" role="UNKNOWN">p</XMTok>
<XMTok font="italic" role="UNKNOWN">W</XMTok>
<XMTok font="italic" role="UNKNOWN">u</XMTok>
<XMTok font="italic" role="UNKNOWN">p</XMTok>
</XMApp>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="right">
<XMApp enclose="box" xml:id="S10.Ex80.m1.36">
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok font="italic" role="UNKNOWN">L</XMTok>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok font="italic" role="UNKNOWN">s</XMTok>
<XMTok font="italic" role="UNKNOWN">t</XMTok>
<XMTok font="italic" role="UNKNOWN">l</XMTok>
<XMTok font="italic" role="UNKNOWN">i</XMTok>
<XMTok font="italic" role="UNKNOWN">n</XMTok>
<XMTok font="italic" role="UNKNOWN">e</XMTok>
</XMApp>
</XMCell>
</XMRow>
</XMArray>
</XMDual>
</XMath>
</Math>
</equation>
</para>
</section>
<section inlist="toc" xml:id="S11">
<tags>
<tag>11</tag>
<tag role="refnum">11</tag>
<tag role="typerefnum">§11</tag>
</tags>
<title><tag close=" ">11</tag>Stepped lines</title>
<para xml:id="S11.p1">
<equation xml:id="S11.Ex81">
<Math class="ltx_math_unparsed" mode="display" tex="\begin{gathered}x=1,\quad x+1=2\\ y=2\end{gathered}" text="[1∗ ]@x@=@1@,@quad@x@+@1@=@2@[ over]@[2∗ ]@y@=@2@[ over]" xml:id="S11.Ex81.m1">
<XMath>
<XMDual>
<XMWrap rule="Anything,">
<XMRef idref="S11.Ex81.m1.1"/>
<XMRef idref="S11.Ex81.m1.2"/>
<XMRef idref="S11.Ex81.m1.3"/>
<XMRef idref="S11.Ex81.m1.4"/>
<XMRef idref="S11.Ex81.m1.5"/>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMRef idref="S11.Ex81.m1.6"/>
<XMRef idref="S11.Ex81.m1.7"/>
<XMRef idref="S11.Ex81.m1.8"/>
<XMRef idref="S11.Ex81.m1.9"/>
<XMRef idref="S11.Ex81.m1.10"/>
<XMRef idref="S11.Ex81.m1.11"/>
<XMRef idref="S11.Ex81.m1.12"/>
<XMRef idref="S11.Ex81.m1.13"/>
<XMRef idref="S11.Ex81.m1.14"/>
<XMRef idref="S11.Ex81.m1.15"/>
<XMRef idref="S11.Ex81.m1.16"/>
</XMWrap>
<XMArray name="stargathered">
<XMRow>
<XMCell align="right">
<XMWrap>
<XMApp xml:id="S11.Ex81.m1.17">
<XMTok meaning="equals" role="RELOP" xml:id="S11.Ex81.m1.3">=</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMText xml:id="S11.Ex81.m1.1"><text width="0.0pt">1</text><Math mode="inline" tex="*" text="*" xml:id="S11.Ex81.m1.m1">
<XMath>
<XMTok meaning="times" role="MULOP">∗</XMTok>
</XMath>
</Math> </XMText>
<XMTok font="italic" role="UNKNOWN" xml:id="S11.Ex81.m1.2">x</XMTok>
</XMApp>
<XMTok meaning="1" role="NUMBER" xml:id="S11.Ex81.m1.4">1</XMTok>
</XMApp>
<XMTok role="PUNCT" rpadding="10.0pt" xml:id="S11.Ex81.m1.5">,</XMTok>
<XMApp xml:id="S11.Ex81.m1.18">
<XMTok meaning="equals" role="RELOP" xml:id="S11.Ex81.m1.9">=</XMTok>
<XMApp>
<XMTok meaning="plus" role="ADDOP" xml:id="S11.Ex81.m1.7">+</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S11.Ex81.m1.6">x</XMTok>
<XMTok meaning="1" role="NUMBER" xml:id="S11.Ex81.m1.8">1</XMTok>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok meaning="2" role="NUMBER" xml:id="S11.Ex81.m1.10">2</XMTok>
<XMText xml:id="S11.Ex81.m1.11"> <text font="bold">over</text></XMText>
</XMApp>
</XMApp>
</XMWrap>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="right">
<XMApp>
<XMTok meaning="equals" role="RELOP" xml:id="S11.Ex81.m1.14">=</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMText xml:id="S11.Ex81.m1.12"><text width="0.0pt">2</text><Math mode="inline" tex="*" text="*" xml:id="S11.Ex81.m1.m2">
<XMath>
<XMTok meaning="times" role="MULOP">∗</XMTok>
</XMath>
</Math> </XMText>
<XMTok font="italic" role="UNKNOWN" xml:id="S11.Ex81.m1.13">y</XMTok>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok meaning="2" role="NUMBER" xml:id="S11.Ex81.m1.15">2</XMTok>
<XMText xml:id="S11.Ex81.m1.16"> <text font="bold">over</text></XMText>
</XMApp>
</XMApp>
</XMCell>
</XMRow>
</XMArray>
</XMDual>
</XMath>
</Math>
</equation>
<p>42</p>
</para>
<para xml:id="S11.p2">
<equation xml:id="S11.Ex82">
<Math class="ltx_math_unparsed" mode="display" tex="\begin{gathered}s=2.8,\quad s+0.2=3\\ t=4.5\end{gathered}" text="[See:]@s@=@2.8@,@quad@s@+@0.2@=@3@[the end]@[See:]@t@=@4.5@[the end]" xml:id="S11.Ex82.m1">
<XMath>
<XMDual>
<XMWrap rule="Anything,">
<XMRef idref="S11.Ex82.m1.1"/>
<XMRef idref="S11.Ex82.m1.2"/>
<XMRef idref="S11.Ex82.m1.3"/>
<XMRef idref="S11.Ex82.m1.4"/>
<XMRef idref="S11.Ex82.m1.5"/>
<XMHint name="quad" role="PUNCT" width="10pt"/>
<XMRef idref="S11.Ex82.m1.6"/>
<XMRef idref="S11.Ex82.m1.7"/>
<XMRef idref="S11.Ex82.m1.8"/>
<XMRef idref="S11.Ex82.m1.9"/>
<XMRef idref="S11.Ex82.m1.10"/>
<XMRef idref="S11.Ex82.m1.11"/>
<XMRef idref="S11.Ex82.m1.12"/>
<XMRef idref="S11.Ex82.m1.13"/>
<XMRef idref="S11.Ex82.m1.14"/>
<XMRef idref="S11.Ex82.m1.15"/>
<XMRef idref="S11.Ex82.m1.16"/>
</XMWrap>
<XMArray name="stargathered">
<XMRow>
<XMCell align="right">
<XMWrap>
<XMApp xml:id="S11.Ex82.m1.17">
<XMTok meaning="equals" role="RELOP" xml:id="S11.Ex82.m1.3">=</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMText xml:id="S11.Ex82.m1.1">See:</XMText>
<XMTok font="italic" role="UNKNOWN" xml:id="S11.Ex82.m1.2">s</XMTok>
</XMApp>
<XMTok meaning="2.8" role="NUMBER" xml:id="S11.Ex82.m1.4">2.8</XMTok>
</XMApp>
<XMTok role="PUNCT" rpadding="10.0pt" xml:id="S11.Ex82.m1.5">,</XMTok>
<XMApp xml:id="S11.Ex82.m1.18">
<XMTok meaning="equals" role="RELOP" xml:id="S11.Ex82.m1.9">=</XMTok>
<XMApp>
<XMTok meaning="plus" role="ADDOP" xml:id="S11.Ex82.m1.7">+</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S11.Ex82.m1.6">s</XMTok>
<XMTok meaning="0.2" role="NUMBER" xml:id="S11.Ex82.m1.8">0.2</XMTok>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok meaning="3" role="NUMBER" xml:id="S11.Ex82.m1.10">3</XMTok>
<XMText xml:id="S11.Ex82.m1.11">the end</XMText>
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</XMApp>
</XMWrap>
</XMCell>
</XMRow>
<XMRow>
<XMCell align="right">
<XMApp>
<XMTok meaning="equals" role="RELOP" xml:id="S11.Ex82.m1.14">=</XMTok>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMText xml:id="S11.Ex82.m1.12">See:</XMText>
<XMTok font="italic" role="UNKNOWN" xml:id="S11.Ex82.m1.13">t</XMTok>
</XMApp>
<XMApp>
<XMTok meaning="times" role="MULOP"></XMTok>
<XMTok meaning="4.5" role="NUMBER" xml:id="S11.Ex82.m1.15">4.5</XMTok>
<XMText xml:id="S11.Ex82.m1.16">the end</XMText>
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</XMApp>
</XMCell>
</XMRow>
</XMArray>
</XMDual>
</XMath>
</Math>
</equation>
<p>1337</p>
</para>
</section>
<section inlist="toc" xml:id="S12">
<tags>
<tag>12</tag>
<tag role="refnum">12</tag>
<tag role="typerefnum">§12</tag>
</tags>
<title><tag close=" ">12</tag>Shifting equations</title>
<para xml:id="S12.p1">
<equationgroup class="ltx_eqn_align" xml:id="S12.EGx10">
<equation xml:id="S12.Ex83">
<MathFork>
<text class="ltx_markedasmath" framecolor="#000000" framed="rectangle" width="284.5pt">Part 1</text>
<MathBranch>
<td/>
<td align="left"><text class="ltx_markedasmath" framecolor="#000000" framed="rectangle" width="284.5pt">Part 1</text></td>
</MathBranch>
</MathFork>
</equation>
<equation xml:id="S12.Ex84">
<MathFork>
<Math tex="\displaystyle={}\framebox[227.62204pt][c]{2nd line}" text="absent = [2nd line]" xml:id="S12.Ex84.m3">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok meaning="absent"/>
<XMText class="ltx_markedasmath" framecolor="#000000" framed="rectangle" width="227.6pt">2nd line</XMText>
</XMApp>
</XMath>
</Math>
<MathBranch>
<td align="right"><Math mode="inline" tex="\displaystyle={}" text="=" xml:id="S12.Ex84.m1">
<XMath>
<XMTok meaning="equals" role="RELOP">=</XMTok>
</XMath>
</Math></td>
<td align="left"><text class="ltx_markedasmath" framecolor="#000000" framed="rectangle" width="227.6pt">2nd line</text></td>
</MathBranch>
</MathFork>
</equation>
<equation xml:id="S12.Ex85">
<MathFork>
<Math tex="\displaystyle 19+\framebox[113.81102pt][c]{ last part}" text="19 + [ last part]" xml:id="S12.Ex85.m3">
<XMath>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMTok meaning="19" role="NUMBER">19</XMTok>
<XMText framecolor="#000000" framed="rectangle" width="113.8pt"> last part</XMText>
</XMApp>
</XMath>
</Math>
<MathBranch>
<td align="right"><Math mode="inline" tex="\displaystyle 19" text="19" xml:id="S12.Ex85.m1">
<XMath>
<XMTok meaning="19" role="NUMBER">19</XMTok>
</XMath>
</Math></td>
<td align="left"><Math mode="inline" tex="\displaystyle+\framebox[113.81102pt][c]{ last part}" text="+ [ last part]" xml:id="S12.Ex85.m2">
<XMath>
<XMApp>
<XMTok meaning="plus" role="ADDOP">+</XMTok>
<XMText framecolor="#000000" framed="rectangle" width="113.8pt"> last part</XMText>
</XMApp>
</XMath>
</Math></td>
</MathBranch>
</MathFork>
</equation>
</equationgroup>
</para>
<para xml:id="S12.p2">
<equationgroup class="ltx_eqn_align" xml:id="S12.EGx11">
<equation xml:id="S12.E27">
<tags>
<tag>(27)</tag>
<tag role="refnum">27</tag>
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<MathFork>
<Math tex="\displaystyle\framebox[42.67912pt]{1}=\framebox[85.35826pt]{2}" text="[1] = [2]" xml:id="S12.E27.m3">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMText class="ltx_markedasmath" framecolor="#000000" framed="rectangle" width="42.7pt">1</XMText>
<XMText framecolor="#000000" framed="rectangle" width="85.4pt">2</XMText>
</XMApp>
</XMath>
</Math>
<MathBranch>
<td align="right"><text class="ltx_markedasmath" framecolor="#000000" framed="rectangle" width="42.7pt">1</text></td>
<td align="left"><Math mode="inline" tex="\displaystyle=\framebox[85.35826pt]{2}" text="absent = [2]" xml:id="S12.E27.m2">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok meaning="absent"/>
<XMText framecolor="#000000" framed="rectangle" width="85.4pt">2</XMText>
</XMApp>
</XMath>
</Math></td>
</MathBranch>
</MathFork>
</equation>
<equation xml:id="S12.E28">
<tags>
<tag>(28)</tag>
<tag role="refnum">28</tag>
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<MathFork>
<Math tex="\displaystyle\framebox[42.67912pt]{3}=\framebox[56.9055pt]{4}" text="[3] = [4]" xml:id="S12.E28.m3">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMText class="ltx_markedasmath" framecolor="#000000" framed="rectangle" width="42.7pt">3</XMText>
<XMText framecolor="#000000" framed="rectangle" width="56.9pt">4</XMText>
</XMApp>
</XMath>
</Math>
<MathBranch>
<td align="right"><text class="ltx_markedasmath" framecolor="#000000" framed="rectangle" width="42.7pt">3</text></td>
<td align="left"><Math mode="inline" tex="\displaystyle=\framebox[56.9055pt]{4}" text="absent = [4]" xml:id="S12.E28.m2">
<XMath>
<XMApp>
<XMTok meaning="equals" role="RELOP">=</XMTok>
<XMTok meaning="absent"/>
<XMText framecolor="#000000" framed="rectangle" width="56.9pt">4</XMText>
</XMApp>
</XMath>
</Math></td>
</MathBranch>
</MathFork>
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