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<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">K</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">V1</font> be  <a href="matrlin.html#V2" target="_self">finite-dimensional</a> <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">M</font> be    <a href="matrix_1.html#M1">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">m</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>;<br/>









<b>assume </b><a NAME="E1:50"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">K</font>, <font color="Maroon">d</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">K</font>;<br/>

<b>assume </b><a NAME="E2:50"/><i><font color="Green">E44</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; ( for <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">d</font> holds <br/><font color="Maroon">d</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Olive">j</font></span>)</span></span>)</span> ) )
 ;<br/>

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">V1</font>, <font color="Maroon">c</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">V1</font>;<br/>

<b>assume </b><a NAME="E3:50"/><i><font color="Green">E50</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c</font> holds <br/><font color="Maroon">c</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" target="_self">lmlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Olive">i</font></span>)</span>,<font color="Maroon">b</font></span>)</span> ) )
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>,  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>2</sub></font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font>, <font color="Maroon">m1</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon">M1</font> being    <a href="matrix_1.html#M1">Matrix</a> of <font color="Maroon">n1</font>,<font color="Maroon">m1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V1</font><b> such that </b><br/><a NAME="E6:50"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M1</font> holds <br/><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>,<font color="Olive">j</font>)
 <b>from </b><i><a class="ref" href="matrix_1.html#S1">MATRIX_1:sch 1</a>();<br/></i>
<a NAME="E7:50"/><i><font color="Green">E54</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a></i>;<br/>
<a NAME="E8:50"/><i><font color="Green">E55</font></i>: 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a></i>;<br/>
<a NAME="E9:50"/><i><font color="Green">E57</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a></i>;<br/>
<a NAME="E10:50"/><i><font color="Green">E58</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E11:50"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" target="_self">lmlt</a> <font color="Maroon">p</font>,<font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<b>then </b><i><font color="Green">E66</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" target="_self">lmlt</a> <font color="Maroon">p</font>,<font color="Maroon">c</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font>
<b>by </b><i>, <a class="ref" href="matrix_1.html#D3">MATRIX_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M1</font></span>)</span>
<b>by </b><i/>
;<br/>
<div><i><font color="Green">E67</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:50_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M1</font></span>)</span>
 ;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V1</font> = <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K7" target="_self">lmlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">b</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">F</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">V1</font><b> such that </b><br/><a NAME="E3:50_2"/><i><font color="Green">E69</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">j</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i><a NAME="E4:50_2"/><i><font color="Green">E111</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font>
 <b>by </b><i>, , , <a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a></i>;<br/><a NAME="E5:50_2"/><i><font color="Green">E112</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font>
 <b>by </b><i>, , , <a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a></i>;<br/><a NAME="E6:50_2"/><i><font color="Green">E113</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E6:50"><i><font color="Green">E52</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><i><font color="Green">E134</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
<b>by </b><i>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E8:50_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrlin.html#E3:50"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E9:50_2"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E10:50_2"/><b>then </b><i><font color="Green">E135</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">M1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><i><font color="Green">E136</font></i>: <font color="Maroon">M1</font> <a href="matrlin.html#K1" target="_self">/.</a> <font color="Maroon">i</font> = 
<font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="matrix_2.html#T18">MATRIX_2:18</a></i>
;<br/><div><i><font color="Green">E137</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:50_2_3"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 ;<br/><a NAME="E2:50_2_3"/><b>then </b><i><font color="Green">E155</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:50_2_3"/><b>then </b><i><font color="Green">E156</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" target="_self">lmlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrlin.html#E8:50"><i><font color="Green">E55</font></i></a>, </i>;<br/><a NAME="E4:50_2_3"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="matrlin.html#T5" target="_self">Th5</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E5:50_2_3"/><i><font color="Green">E160</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 <b>by </b><i>, , <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E6:50_2_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">M1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="matrlin.html#E6:50"><i><font color="Green">E52</font></i></a>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E7:50_2_3"/><b>then </b><i><font color="Green">E161</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D5">MATRIX_1:def 5</a></i>;<br/><i><font color="Green">E162</font></i>: <span class="p1">(<span class="default"><a href="matrlin.html#K7" target="_self">lmlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">b</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><a href="matrlin.html#K7" target="_self">lmlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="matrlin.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V1</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="matrlin.html#T6" target="_self">Th6</a>, <a class="ref" href="funcop_1.html#T28">FUNCOP_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="vectsp_1.html#D24">VECTSP_1:def 24</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K1" target="_self">/.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
<font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="matrlin.html#E6:50"><i><font color="Green">E52</font></i></a>, <a class="txt" href="matrlin.html#E9:50"><i><font color="Green">E57</font></i></a>, , <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K7" target="_self">lmlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">b</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E11:50"><i><font color="Green">E65</font></i></a>, <a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div><a NAME="E13:50_2"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="matrlin.html#T3" target="_self">Th3</a>, <a class="ref" href="matrlin.html#T4" target="_self">Th4</a>, , <a class="txt" href="matrlin.html#E3:50"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E14:50_2"/><b>then </b><i><font color="Green">E164</font></i>: 
<font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E15:50_2"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="matrlin.html#T3" target="_self">Th3</a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E16:50_2"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V1</font> <a href="finseqop.html#K1">.:</a> <font color="Maroon">p</font>,<font color="Maroon">c</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="matrlin.html#E3:50"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E17:50_2"/><i><font color="Green">E167</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" target="_self">lmlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E8:50"><i><font color="Green">E55</font></i></a>, </i>;<br/><a NAME="E18:50_2"/><b>then </b><i><font color="Green">E168</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" target="_self">lmlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E19:50_2"/><i><font color="Green">E169</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" target="_self">lmlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><div><i><font color="Green">E170</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:50_2_4"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E2:50_2_4"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E3:50_2_4"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K7" target="_self">lmlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">b</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>thus </b><span class="p1">(<span class="default"><a href="matrlin.html#K7" target="_self">lmlt</a> <font color="Maroon">p</font>,<font color="Maroon">c</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V1</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T28">FUNCOP_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="matrlin.html#T5" target="_self">Th5</a>, <a class="ref" href="matrlin.html#T6" target="_self">Th6</a>, <a class="ref" href="vectsp_1.html#D24">VECTSP_1:def 24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><a href="matrlin.html#K7" target="_self">lmlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">F</font>
<b>by </b><i>, <a class="txt" href="matrlin.html#E11:50"><i><font color="Green">E65</font></i></a>, <a class="txt" href="matrlin.html#E12:50"><i><font color="Green">E66</font></i></a>, <a class="ref" href="vectsp_3.html#T10">VECTSP_3:10</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K1" target="_self">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E7:50"><i><font color="Green">E54</font></i></a>, <a class="txt" href="matrlin.html#E9:50"><i><font color="Green">E57</font></i></a>, <a class="txt" href="matrlin.html#E10:50"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="txt" href="matrlin.html#E3:50"><i><font color="Green">E50</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="txt" href="matrlin.html#E3:50"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E14:50"/><i><font color="Green">E171</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E15:50"/><b>then </b><i><font color="Green">E172</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" target="_self">lmlt</a> <font color="Maroon">d</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">d</font>
 
<b>by </b><i/>;<br/>
<a NAME="E16:50"/><i><font color="Green">E174</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" target="_self">lmlt</a> <font color="Maroon">d</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="txt" href="matrlin.html#E3:50"><i><font color="Green">E50</font></i></a>, </i>;<br/>
<i><font color="Green">E175</font></i>:  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<i><font color="Green">E178</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" target="_self">lmlt</a> <font color="Maroon">d</font>,<font color="Maroon">b</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="matrlin.html#E6:50"><i><font color="Green">E52</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span>
<b>by </b><i>, , , <a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a></i>
;<br/>
<div><i><font color="Green">E179</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:50_5"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:50_5"/><i><font color="Green">E181</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">H</font> =  <a href="fvsum_1.html#K11">mlt</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">j</font></span>)</span> as   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">K</font> ;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V1</font> = <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>;<br/><b>consider </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">V1</font><b> such that </b><br/><a NAME="E5:50_5"/><i><font color="Green">E182</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> holds <br/><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">i</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i><a NAME="E6:50_5"/><i><font color="Green">E183</font></i>: 
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> holds <br/><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">i</font>)
 <b>by </b><i><a class="txt" href="matrlin.html#E10:50"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E7:50_5"/><i><font color="Green">E184</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E8:50_5"/><i><font color="Green">E185</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E10:50"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><i><font color="Green">E186</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">j</font></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font>
<b>by </b><i>, , <a class="txt" href="matrlin.html#E1:50"><i><font color="Green">E43</font></i></a>, <a class="txt" href="matrlin.html#E2:50"><i><font color="Green">E44</font></i></a>, <a class="txt" href="matrlin.html#E10:50"><i><font color="Green">E58</font></i></a>, <a class="ref" href="matrix_2.html#T15">MATRIX_2:15</a></i>
;<br/><a NAME="E10:50_5"/><i><font color="Green">E187</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E11:50_5"/><b>then </b><i><font color="Green">E188</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finseqop.html#K1">.:</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T86">FINSEQ_2:86</a></i>;<br/><div><i><font color="Green">E189</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_5_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:50_5_2"/><i><font color="Green">E190</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span>
 ;<br/><a NAME="E2:50_5_2"/><b>then </b><i><font color="Green">E191</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E3:50_5_2"/><i><font color="Green">E192</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:50_5_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="txt" href="matrlin.html#E1:50"><i><font color="Green">E43</font></i></a>, <a class="txt" href="matrlin.html#E10:50"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E5:50_5_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">M1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, , <a class="txt" href="matrlin.html#E12:50"><i><font color="Green">E66</font></i></a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E6:50_5_2"/><b>then </b><i><font color="Green">E193</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D5">MATRIX_1:def 5</a></i>;<br/><a NAME="E7:50_5_2"/><i><font color="Green">E194</font></i>: 
<font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="matrlin.html#T6" target="_self">Th6</a>, <a class="txt" href="matrlin.html#E12:50"><i><font color="Green">E66</font></i></a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E8:50_5_2"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finseqop.html#K1">.:</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrlin.html#E10:50"><i><font color="Green">E58</font></i></a>, , <a class="txt" href="matrlin.html#E12:50"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_2.html#T86">FINSEQ_2:86</a></i>;<br/><a NAME="E9:50_5_2"/><b>then </b><i><font color="Green">E196</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E10:50_5_2"/><i><font color="Green">E197</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finseqop.html#K1">.:</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="matrlin.html#D1" target="_self">Def1</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E11:50_5_2"/><i><font color="Green">E198</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><i><font color="Green">E199</font></i>: <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> = 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i>, <a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , <a class="txt" href="matrlin.html#E10:50"><i><font color="Green">E58</font></i></a>, <a class="txt" href="matrlin.html#E12:50"><i><font color="Green">E66</font></i></a>, , <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finseqop.html#K1">.:</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T28">FUNCOP_1:28</a></i>
<br/>.= 
<font color="Maroon">H</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font>
<b>by </b><i><a class="txt" href="matrlin.html#E13:50"><i><font color="Green">E67</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
<font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
<b>by </b><i>, <a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="matrlin.html#D2" target="_self">Def2</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="txt" href="matrlin.html#E10:50"><i><font color="Green">E58</font></i></a>, <a class="txt" href="matrlin.html#E12:50"><i><font color="Green">E66</font></i></a></i>
;<br/></div><b>end;</b></div><a NAME="E13:50_5"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E14:50_5"/><i><font color="Green">E201</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" target="_self">lmlt</a> <font color="Maroon">d</font>,<font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrlin.html#E7:50"><i><font color="Green">E54</font></i></a>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E15:50_5"/><b>then </b><i><font color="Green">E202</font></i>: 
<font color="Maroon">d</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E16:50_5"/><i><font color="Green">E203</font></i>: 
<font color="Maroon">b</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E6:50"><i><font color="Green">E52</font></i></a>, <a class="txt" href="matrlin.html#E13:50"><i><font color="Green">E67</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E17:50_5"/><i><font color="Green">E204</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E3:50"><i><font color="Green">E50</font></i></a>, <a class="txt" href="matrlin.html#E13:50"><i><font color="Green">E67</font></i></a>, </i>;<br/><a NAME="E18:50_5"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E10:50"><i><font color="Green">E58</font></i></a>, </i>;<br/><a NAME="E19:50_5"/><b>then </b><i><font color="Green">E205</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E20:50_5"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span>
 <b>by </b><i/>;<br/><a NAME="E21:50_5"/><b>then </b><i><font color="Green">E206</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><b>then </b><i><font color="Green">E207</font></i>: <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrlin.html#K1" target="_self">/.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="matrlin.html#E12:50"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="matrix_1.html#K8">Line</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="matrlin.html#E12:50"><i><font color="Green">E66</font></i></a>, , <a class="ref" href="matrix_2.html#T18">MATRIX_2:18</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><a href="matrlin.html#K7" target="_self">lmlt</a> <font color="Maroon">d</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V1</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E13:50"><i><font color="Green">E67</font></i></a>, <a class="ref" href="funcop_1.html#T28">FUNCOP_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="vectsp_1.html#D24">VECTSP_1:def 24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">H</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">G</font>
<b>by </b><i>, <a class="txt" href="matrlin.html#E10:50"><i><font color="Green">E58</font></i></a>, <a class="ref" href="matrlin.html#T5" target="_self">Th5</a>, <a class="ref" href="matrlin.html#T6" target="_self">Th6</a>, , <a class="ref" href="matrlin.html#D1" target="_self">Def1</a>, </i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrlin.html#K1" target="_self">/.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, <a class="txt" href="matrlin.html#E11:50"><i><font color="Green">E65</font></i></a>, , <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="matrlin.html#E12:50"><i><font color="Green">E66</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, <a class="txt" href="matrlin.html#E9:50"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/></div><b>end;</b></div>
<b>thus </b> <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" target="_self">lmlt</a> <font color="Maroon">p</font>,<font color="Maroon">c</font></span>)</span> = 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <font color="Maroon">M1</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="newton.html#T3">NEWTON:3</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="matrlin.html#K7" target="_self">lmlt</a> <font color="Maroon">d</font>,<font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E7:50"><i><font color="Green">E54</font></i></a>, <a class="txt" href="matrlin.html#E8:50"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>
;<br/>


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