<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">K</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="matrix_1.html#NM4">Matrix</a> of <font color="Maroon">K</font>, <font color="Maroon">B</font> be   <a href="matrix_1.html#NM4">Matrix</a> of <font color="Maroon">K</font>, <font color="Maroon">C</font> be   <a href="matrix_1.html#NM4">Matrix</a> of <font color="Maroon">K</font>;<br/>



<b>assume </b><a NAME="E1:43"/><i><font color="Green">E43</font></i>: 
(  <a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> )
 ;<br/>

<a NAME="E2:43"/><i><font color="Green">E44</font></i>: 
(  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">A</font> &amp;  <a href="setwop_2.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> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">B</font> &amp;  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:43"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font>
 
<b>by </b><i><a class="ref" href="fvsum_1.html#T8">FVSUM_1:8</a></i>;<br/>
<a NAME="E4:43"/><b>then </b><i><font color="Green">E45</font></i>: 
( the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> is <a href="binop_1.html#V1">commutative</a> &amp; the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> is <a href="binop_1.html#V2">associative</a> &amp; the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  &amp; the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="finseqop.html#NR2" target="_self">has_an_inverseOp</a>  )
 
<b>by </b><i><a class="ref" href="fvsum_1.html#T17">FVSUM_1:17</a>, <a class="ref" href="setwiseo.html#D2">SETWISEO:def 2</a></i>;<br/>
<a NAME="E5:43"/><i><font color="Green">E48</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:43"/><i><font color="Green">E49</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">C</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:43"/><b>then </b><i><font color="Green">E50</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">C</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E8:43"/><i><font color="Green">E51</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">C</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:43"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E10:43"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">C</font></span>)</span></span>)</span></span><a href="matrix_3.html#K8" target="_self">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="matrix_1.html#D5">MATRIX_1:def 5</a></i>;<br/>
<a NAME="E11:43"/>
 <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E12:43"/><b>then </b><i><font color="Green">E68</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">B</font></span>)</span></span>)</span></span><a href="matrix_3.html#K8" target="_self">:]</a></span>
 
<b>by </b><i>, , <a class="ref" href="matrix_1.html#D5">MATRIX_1:def 5</a></i>;<br/>
<a NAME="E13:43"/>
 <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">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="E14:43"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">C</font></span>)</span></span>)</span></span><a href="matrix_3.html#K8" target="_self">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="matrix_1.html#D5">MATRIX_1:def 5</a></i>;<br/>
<a NAME="E15:43"/>
 <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E16:43"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">C</font></span>)</span></span>)</span></span><a href="matrix_3.html#K8" target="_self">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="matrix_1.html#D5">MATRIX_1:def 5</a></i>;<br/>
<a NAME="E17:43"/><i><font color="Green">E76</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">C</font></span>)</span></span>)</span></span><a href="matrix_3.html#K8" target="_self">:]</a></span>
 
<b>by </b><i>, , <a class="ref" href="matrix_1.html#D5">MATRIX_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:43_1"/><i><font color="Green">E77</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span>
 ;<br/><a NAME="E2:43_1"/><b>then </b><i><font color="Green">E78</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">A</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">C</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/> <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">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">i</font></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></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> <font color="Maroon">C</font>
<b>by </b><i>, </i>
<br/>.= 
 <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">C</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/><b>then </b><i><font color="Green">E79</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> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span> = 
 <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">C</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T86">FINSEQ_2:86</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
;<br/><a NAME="E5:43_1"/><i><font color="Green">E80</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> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="fvsum_1.html#D7">FVSUM_1:def 7</a></i>;<br/><b>reconsider </b><font color="Maroon">X</font> =  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <a href="numbers.html#K5">NAT</a>  ;<br/><b>reconsider </b><font color="Maroon">Y</font> =  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">B</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <a href="numbers.html#K5">NAT</a>  ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">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">K</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">a<sub>2</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">j</font></span>)</span>;<br/><b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font><b> such that </b><br/><a NAME="E9:43_1"/><i><font color="Green">E81</font></i>: 
for <font color="Olive">k1</font>, <font color="Olive">k2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">k1</font>,<font color="Olive">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k1</font>,<font color="Olive">k2</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4">BINOP_1:sch 4</a>();<br/></i><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">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">K</font> = the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">Y</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">f</font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font><b> such that </b><br/><a NAME="E11:43_1"/><i><font color="Green">E82</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">k</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i><b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font> = the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3">curry'</a> <font color="Maroon">f</font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">h</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font><b> such that </b><br/><a NAME="E13:43_1"/><i><font color="Green">E83</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">k</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i><a NAME="E14:43_1"/><i><font color="Green">E84</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="fvsum_1.html#T9">FVSUM_1:9</a></i>;<br/><a NAME="E15:43_1"/>
 <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="txt" href="matrix_3.html#E1"><i><font color="Green">Lemma38</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E16:43_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default"><a href="setwop_2.html#K1">[#]</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p4">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fvsum_1.html#K11">mlt</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="setwop_2.html#T23">SETWOP_2:23</a></i>;<br/><a NAME="E17:43_1"/><i><font color="Green">E86</font></i>: 
 <a href="setwop_2.html#K3">dom</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:43_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:43_1_3"/><b>then </b>
<font color="Maroon">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="setwiseo.html#T14">SETWISEO:14</a></i>;<br/>
<b>hence </b><a NAME="E3:43_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E18:43_1"/><i><font color="Green">E87</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">f</font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Olive">k</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">k</font>,<font color="Maroon">j</font></span>)</span> <a href="fvsum_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Olive">k</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:43_1_4"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<a NAME="E2:43_1_4"/><i><font color="Green">E88</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">k</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E3:43_1_4"/><i><font color="Green">E89</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:43_1_4"/><i><font color="Green">E90</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green">E91</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">f</font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Maroon">k</font></span>)</span> = 
 <a href="funct_5.html#K2">proj2</a> <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">k</font></span><a href="tarski.html#K1">}</a></span>,<span class="p3">(<span class="default"><a href="funct_5.html#K2">proj2</a> <span class="p4"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_5.html#T38">FUNCT_5:38</a></i>
<br/>.= 
 <a href="funct_5.html#K2">proj2</a> <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">k</font></span><a href="tarski.html#K1">}</a></span>,<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="funct_5.html#T11">FUNCT_5:11</a></i>
<br/>.= 
 <a href="funct_5.html#K2">proj2</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">k</font></span><a href="tarski.html#K1">}</a></span>,<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T124">ZFMISC_1:124</a></i>
<br/>.= 
 <a href="numbers.html#K5">NAT</a> 
<b>by </b><i><a class="ref" href="funct_5.html#T11">FUNCT_5:11</a></i>
;<br/>
<i><font color="Green">E92</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">A</font>,<font color="Maroon">i</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font>
<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"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
;<br/>
<a NAME="E7:43_1_4"/><i><font color="Green">E118</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <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> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="fvsum_1.html#D7">FVSUM_1:def 7</a></i>;<br/>
<i><font color="Green">E119</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> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span></span>)</span> = 
 <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">A</font>,<font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_3.html#E7:43"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_2.html#T86">FINSEQ_2:86</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font>
<b>by </b><i>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/>
<a NAME="E9:43_1_4"/><i><font color="Green">E120</font></i>: 
 <a href="setwop_2.html#K3">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> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.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> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span></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">a</font> = <font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font> ;<br/>
<i><font color="Green">E121</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">f</font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">B</font></span>)</span></span>)</span> = 
<a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_3.html#E6:43"><i><font color="Green">E49</font></i></a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
 <a href="setwop_2.html#K3">dom</a> <font color="Maroon">B</font>
<b>by </b><i><a class="txt" href="matrix_3.html#E2:43"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">p</font> =  <a href="fvsum_1.html#K11">mlt</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span> as   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">K</font> ;<br/>
<a NAME="E13:43_1_4"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">a</font> <a href="fvsum_1.html#K9">*</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="fvsum_1.html#K1">multfield</a> </span>)</span> <a href="finseqop.html#K5">*</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="fvsum_1.html#D6">FVSUM_1:def 6</a></i>;<br/>
<a NAME="E14:43_1_4"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="fvsum_1.html#K1">multfield</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_4_5"><b>proof </b></a><div class="add">

<a NAME="E1:43_1_4_5"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="fvsum_1.html#K1">multfield</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:43_1_4_5"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="fvsum_1.html#K1">multfield</a> </span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:43_1_4"/><b>then </b><i><font color="Green">E138</font></i>: 
 <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="fvsum_1.html#K9">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="matrix_3.html#T3" target="_self">Th3</a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E16:43_1_4"/><b>then </b><i><font color="Green">E139</font></i>: 
 <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span> <a href="fvsum_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="txt" href="matrix_3.html#E8:43"><i><font color="Green">E51</font></i></a>, <a class="txt" href="matrix_3.html#E9:43"><i><font color="Green">E52</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_4_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">l</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:43_1_4_6"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">B</font>
 ;<br/><b>then reconsider </b><font color="Maroon">l'</font> = <font color="Maroon">l</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="setwiseo.html#T14">SETWISEO:14</a></i>;<br/><a NAME="E3:43_1_4_6"/>
<font color="Maroon">l'</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="matrix_3.html#E8:43"><i><font color="Green">E51</font></i></a>, <a class="txt" href="matrix_3.html#E9:43"><i><font color="Green">E52</font></i></a>, <a class="ref" href="matrix_3.html#D4" target="_self">Def4</a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font> <b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/><a NAME="E5:43_1_4_6"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">l'</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">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> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix_3.html#E9:43"><i><font color="Green">E52</font></i></a>, , <a class="ref" href="matrix_3.html#D4" target="_self">Def4</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E6:43_1_4_6"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="fvsum_1.html#K9">*</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix_3.html#E8:43"><i><font color="Green">E51</font></i></a>, <a class="txt" href="matrix_3.html#E9:43"><i><font color="Green">E52</font></i></a>, , <a class="ref" href="matrix_3.html#D4" target="_self">Def4</a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">f</font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">f</font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">l'</font>
<b>by </b><i><a class="ref" href="matrix_3.html#D4" target="_self">Def4</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">k</font>,<font color="Maroon">l'</font>
<b>by </b><i><a class="txt" href="matrix_3.html#E5:43"><i><font color="Green">E48</font></i></a>, <a class="txt" href="matrix_3.html#E6:43"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_5.html#T38">FUNCT_5:38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">l'</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">l'</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i/>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">l'</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">l'</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</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#K2">.</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">l'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">l'</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</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">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">l'</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="matrix_3.html#D4" target="_self">Def4</a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</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">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l'</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l'</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#D4" target="_self">Def4</a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</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"><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> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_3.html#E10:43"><i><font color="Green">E59</font></i></a>, <a class="ref" href="funcop_1.html#T28">FUNCOP_1:28</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"><span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#D7">FVSUM_1:def 7</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span> <a href="fvsum_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font>
<b>by </b><i>, <a class="ref" href="fvsum_1.html#T62">FVSUM_1:62</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E18:43_1_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">f</font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span> <a href="fvsum_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="matrix_3.html#D3" target="_self">Def3</a>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E19:43_1"/><i><font color="Green">E144</font></i>: 
 <a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:43_1_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E2:43_1_5"/><b>then </b>
<font color="Maroon">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="setwiseo.html#T14">SETWISEO:14</a></i>;<br/>
<b>hence </b><a NAME="E3:43_1_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E20:43_1"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fvsum_1.html#K11">mlt</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_6"><b>proof </b></a><div class="add">

<i><font color="Green">E146</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
<a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E2:43_1_6"/><i><font color="Green">E147</font></i>: 
 <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="txt" href="matrix_3.html#E1"><i><font color="Green">Lemma38</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_6_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k'</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:43_1_6_2"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">k'</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font>
 ;<br/><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k'</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="setwiseo.html#T14">SETWISEO:14</a></i>;<br/><a NAME="E3:43_1_6_2"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">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> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="matrix_3.html#E7:43"><i><font color="Green">E50</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="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font>
<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"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
;<br/><b>then </b><i><font color="Green">E151</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> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span></span>)</span> = 
 <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">A</font>,<font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T86">FINSEQ_2:86</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font>
<b>by </b><i>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font> ;<br/><b>reconsider </b><font color="Maroon">p</font> =  <a href="fvsum_1.html#K11">mlt</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span> as   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">K</font> ;<br/><a NAME="E8:43_1_6_2"/><i><font color="Green">E152</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font> <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="txt" href="matrix_3.html#E9:43"><i><font color="Green">E52</font></i></a>, <a class="ref" href="fvsum_1.html#D7">FVSUM_1:def 7</a></i>;<br/><a NAME="E9:43_1_6_2"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">a</font> <a href="fvsum_1.html#K9">*</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="fvsum_1.html#K1">multfield</a> </span>)</span> <a href="finseqop.html#K5">*</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="fvsum_1.html#D6">FVSUM_1:def 6</a></i>;<br/><a NAME="E10:43_1_6_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="fvsum_1.html#K1">multfield</a> </span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_6_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:43_1_6_2_3"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="fvsum_1.html#K1">multfield</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:43_1_6_2_3"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="fvsum_1.html#K1">multfield</a> </span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:43_1_6_2"/><b>then </b>
 <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="fvsum_1.html#K9">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="matrix_3.html#D3" target="_self">Def3</a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/><a NAME="E12:43_1_6_2"/><b>then </b><i><font color="Green">E154</font></i>: 
 <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span> <a href="fvsum_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">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="E13:43_1_6_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="setwop_2.html#K1">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span> <a href="fvsum_1.html#K9">*</a> <span class="p3">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p4">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span> <a href="fvsum_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="setwop_2.html#T23">SETWOP_2:23</a></i>;<br/><a NAME="E14:43_1_6_2"/><b>then </b><i><font color="Green">E155</font></i>: 
<span class="p1">(<span class="default"><a href="setwop_2.html#K1">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span> <a href="fvsum_1.html#K9">*</a> <span class="p3">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p4">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">f</font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix_3.html#E4:43"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E15:43_1_6_2"/><i><font color="Green">E156</font></i>: 
( the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> is <a href="binop_1.html#V1">commutative</a> &amp; the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> is <a href="binop_1.html#V2">associative</a> &amp; the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  )
 <b>by </b><i><a class="ref" href="fvsum_1.html#T10">FVSUM_1:10</a></i>;<br/><a NAME="E16:43_1_6_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="matrix_3.html#E7:43"><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="E17:43_1_6_2"/><b>then </b><i><font color="Green">E157</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">k</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E18:43_1_6_2"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.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">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="matrix_3.html#E7:43"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k'</font> = 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="matrix_3.html#E5:43"><i><font color="Green">E48</font></i></a>, <a class="txt" href="matrix_3.html#E7:43"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">Y</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">f</font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#D2" target="_self">Def2</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">Y</font>,<span class="p1">(<span class="default"><a href="setwop_2.html#K1">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span> <a href="fvsum_1.html#K9">*</a> <span class="p3">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p4">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="setwop_2.html#T9">SETWOP_2:9</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="fvsum_1.html#K9">*</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="matrix_3.html#T3" target="_self">Th3</a>, <a class="ref" href="setwop_2.html#D2">SETWOP_2:def 2</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="fvsum_1.html#K9">*</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#D8">FVSUM_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#T92">FVSUM_1:92</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="fvsum_1.html#K13">"*"</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#D10">FVSUM_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i>, <a class="ref" href="matrix_3.html#D4" target="_self">Def4</a>, </i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</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"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_3.html#E10:43"><i><font color="Green">E59</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</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"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</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">C</font>,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_3.html#E7:43"><i><font color="Green">E50</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> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="matrix_3.html#E8:43"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funcop_1.html#T28">FUNCOP_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k'</font>
<b>by </b><i><a class="ref" href="fvsum_1.html#D7">FVSUM_1:def 7</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:43_1_6"/>
<font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fvsum_1.html#K11">mlt</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix_3.html#E5:43"><i><font color="Green">E48</font></i></a>, <a class="txt" href="matrix_3.html#E6:43"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">j</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font>
<b>by </b><i>, </i>
<br/>.= 
 <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">A</font>,<font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/><b>then </b><i><font color="Green">E159</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> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> = 
 <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">A</font>,<font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T86">FINSEQ_2:86</a></i>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/><a NAME="E23:43_1"/><i><font color="Green">E160</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> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="fvsum_1.html#D7">FVSUM_1:def 7</a></i>;<br/><a NAME="E24:43_1"/><b>then </b>
 <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="txt" href="matrix_3.html#E5:43"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E25:43_1"/><b>then </b><i><font color="Green">E161</font></i>: 
<span class="p1">(<span class="default"><a href="setwop_2.html#K1">[#]</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p4">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fvsum_1.html#K11">mlt</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="setwop_2.html#T23">SETWOP_2:23</a></i>;<br/><a NAME="E26:43_1"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">h</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fvsum_1.html#K11">mlt</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">j</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_9"><b>proof </b></a><div class="add">

<a NAME="E1:43_1_9"/><i><font color="Green">E163</font></i>: 
 <a href="setwop_2.html#K3">dom</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:43_1_9_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:43_1_9_1"/><b>then </b>
<font color="Maroon">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="setwiseo.html#T14">SETWISEO:14</a></i>;<br/>
<b>hence </b><a NAME="E3:43_1_9_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>


</div><b>end;</b></div>
<i><font color="Green">E164</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">B</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
<a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="setwop_2.html#K3">dom</a> <font color="Maroon">B</font>
<b>by </b><i><a class="txt" href="matrix_3.html#E9:43"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E3:43_1_9"/><i><font color="Green">E165</font></i>: 
 <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="txt" href="matrix_3.html#E5:43"><i><font color="Green">E48</font></i></a>, <a class="txt" href="matrix_3.html#E6:43"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_9_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k'</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:43_1_9_3"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">k'</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">B</font>
 ;<br/><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k'</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="setwiseo.html#T14">SETWISEO:14</a></i>;<br/><a NAME="E3:43_1_9_3"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">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> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="matrix_3.html#E5:43"><i><font color="Green">E48</font></i></a>, <a class="ref" href="matrix_3.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><i><font color="Green">E168</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">B</font>,<font color="Maroon">k</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font>
<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"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</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="E5:43_1_9_3"/><i><font color="Green">E169</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <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> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="fvsum_1.html#D7">FVSUM_1:def 7</a></i>;<br/><i><font color="Green">E170</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> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span> = 
 <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">B</font>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_2.html#T86">FINSEQ_2:86</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font>
<b>by </b><i>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/><a NAME="E7:43_1_9_3"/>
( <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">G</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:43_1_9_3"/><i><font color="Green">E171</font></i>: 
( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cat_2.html#K3">curry'</a> <font color="Maroon">f</font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Maroon">k</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Maroon">k</font> ) )
 <b>by </b><i><a class="ref" href="funct_5.html#T39">FUNCT_5:39</a></i>;<br/><i><font color="Green">E172</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K3">curry'</a> <font color="Maroon">f</font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font></span>)</span></span>)</span> = 
<a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font></span>)</span>
<b>by </b><i>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
 <a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font> ;<br/><b>reconsider </b><font color="Maroon">p</font> =  <a href="fvsum_1.html#K11">mlt</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span> as   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">K</font> ;<br/><a NAME="E13:43_1_9_3"/><i><font color="Green">E173</font></i>: 
<font color="Maroon">a</font> <a href="fvsum_1.html#K9">*</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="fvsum_1.html#K1">multfield</a> </span>)</span> <a href="finseqop.html#K5">*</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="fvsum_1.html#D6">FVSUM_1:def 6</a></i>;<br/><a NAME="E14:43_1_9_3"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="fvsum_1.html#K1">multfield</a> </span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_9_3_4"><b>proof </b></a><div class="add">

<a NAME="E1:43_1_9_3_4"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="fvsum_1.html#K1">multfield</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:43_1_9_3_4"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="fvsum_1.html#K1">multfield</a> </span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E15:43_1_9_3"/><b>then </b><i><font color="Green">E174</font></i>: 
 <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="fvsum_1.html#K9">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/><a NAME="E16:43_1_9_3"/><b>then </b><i><font color="Green">E175</font></i>: 
 <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font></span>)</span> <a href="fvsum_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="matrix_3.html#D4" target="_self">Def4</a>, <a class="txt" href="matrix_3.html#E10:43"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E17:43_1_9_3"/><b>then </b><i><font color="Green">E176</font></i>: 
<span class="p1">(<span class="default"><a href="setwop_2.html#K1">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font></span>)</span> <a href="fvsum_1.html#K9">*</a> <span class="p3">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span>,<span class="p4">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font></span>)</span> <a href="fvsum_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="setwop_2.html#T23">SETWOP_2:23</a></i>;<br/><a NAME="E18:43_1_9_3"/><i><font color="Green">E177</font></i>: 
<span class="p1">(<span class="default"><a href="setwop_2.html#K1">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font></span>)</span> <a href="fvsum_1.html#K9">*</a> <span class="p3">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span>,<span class="p4">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3">curry'</a> <font color="Maroon">f</font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_9_3_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_9_3_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">l</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:43_1_9_3_5_1"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font>
 ;<br/><b>then reconsider </b><font color="Maroon">l'</font> = <font color="Maroon">l</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="setwiseo.html#T14">SETWISEO:14</a></i>;<br/><a NAME="E3:43_1_9_3_5_1"/>
<font color="Maroon">l'</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="matrix_3.html#D4" target="_self">Def4</a>, <a class="txt" href="matrix_3.html#E10:43"><i><font color="Green">E59</font></i></a>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font> <b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/><a NAME="E5:43_1_9_3_5_1"/><i><font color="Green">E179</font></i>: 
<font color="Maroon">l'</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E6:43_1_9_3_5_1"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">l'</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">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> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix_3.html#E10:43"><i><font color="Green">E59</font></i></a>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E7:43_1_9_3_5_1"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="fvsum_1.html#K9">*</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrix_3.html#D4" target="_self">Def4</a>, <a class="txt" href="matrix_3.html#E10:43"><i><font color="Green">E59</font></i></a>, , , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K3">curry'</a> <font color="Maroon">f</font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3">curry'</a> <font color="Maroon">f</font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">l'</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">l'</font>,<font color="Maroon">k</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">l'</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">l'</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">l'</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">l'</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">l'</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">l'</font>,<font color="Maroon">j</font></span>)</span></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#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">l'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">l'</font>,<font color="Maroon">j</font></span>)</span></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">A</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">l'</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix_3.html#E12:43"><i><font color="Green">E68</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</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">A</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l'</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l'</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</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">A</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><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> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l'</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_3.html#E14:43"><i><font color="Green">E69</font></i></a>, <a class="ref" href="funcop_1.html#T28">FUNCOP_1:28</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">A</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l'</font></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#D7">FVSUM_1:def 7</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font></span>)</span> <a href="fvsum_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font>
<b>by </b><i><a class="ref" href="matrix_3.html#T13" target="_self">Th13</a>, <a class="ref" href="fvsum_1.html#T62">FVSUM_1:62</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:43_1_9_3_5"/>
<span class="p1">(<span class="default"><a href="setwop_2.html#K1">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font></span>)</span> <a href="fvsum_1.html#K9">*</a> <span class="p3">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span>,<span class="p4">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3">curry'</a> <font color="Maroon">f</font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="matrix_3.html#E20"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E19:43_1_9_3"/><i><font color="Green">E182</font></i>: 
( the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> is <a href="binop_1.html#V1">commutative</a> &amp; the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> is <a href="binop_1.html#V2">associative</a> &amp; the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  )
 <b>by </b><i><a class="ref" href="fvsum_1.html#T10">FVSUM_1:10</a></i>;<br/><a NAME="E20:43_1_9_3"/><i><font color="Green">E183</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k</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> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="matrix_3.html#T3" target="_self">Th3</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E21:43_1_9_3"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="matrix_3.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E22:43_1_9_3"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="matrix_3.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k'</font> = 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="matrix_3.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3">curry'</a> <font color="Maroon">f</font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i/>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="setwop_2.html#K1">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font></span>)</span> <a href="fvsum_1.html#K9">*</a> <span class="p3">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span>,<span class="p4">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="setwop_2.html#T9">SETWOP_2:9</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="fvsum_1.html#K9">*</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="setwop_2.html#D2">SETWOP_2:def 2</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="fvsum_1.html#K9">*</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#D8">FVSUM_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#T92">FVSUM_1:92</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">B</font>,<font color="Maroon">k</font></span>)</span> <a href="fvsum_1.html#K13">"*"</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#D10">FVSUM_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i>, <a class="txt" href="matrix_3.html#E12:43"><i><font color="Green">E68</font></i></a>, </i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</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"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_3.html#E14:43"><i><font color="Green">E69</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</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"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#T13" target="_self">Th13</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> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T28">FUNCOP_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k'</font>
<b>by </b><i><a class="ref" href="fvsum_1.html#D7">FVSUM_1:def 7</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:43_1_9"/>
<font color="Maroon">h</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fvsum_1.html#K11">mlt</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="matrix_3.html#D3" target="_self">Def3</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E27:43_1"/><i><font color="Green">E186</font></i>: 
 <a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="matrix_3.html#E1"><i><font color="Green">Lemma38</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E28:43_1"/><i><font color="Green">E187</font></i>: 
 <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="txt" href="matrix_3.html#E5:43"><i><font color="Green">E48</font></i></a>, <a class="txt" href="matrix_3.html#E6:43"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="fvsum_1.html#K13">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#D10">FVSUM_1:def 10</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#D8">FVSUM_1:def 8</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="setwop_2.html#K1">[#]</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p4">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="txt" href="matrix_3.html#E9:43"><i><font color="Green">E52</font></i></a>, <a class="ref" href="setwop_2.html#D2">SETWOP_2:def 2</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">X</font>,<font color="Maroon">g</font>
<b>by </b><i>, <a class="txt" href="matrix_3.html#E1:43"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="setwop_2.html#T9">SETWOP_2:9</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="matrix_3.html#K8" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="matrix_3.html#K8" target="_self">:]</a></span>,<font color="Maroon">f</font>
<b>by </b><i>, <a class="ref" href="matrix_3.html#D2" target="_self">Def2</a>, </i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">Y</font>,<font color="Maroon">h</font>
<b>by </b><i>, , </i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3">dom</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p4">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="setwop_2.html#K1">[#]</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p4">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="txt" href="matrix_3.html#E7:43"><i><font color="Green">E50</font></i></a>, <a class="txt" href="matrix_3.html#E8:43"><i><font color="Green">E51</font></i></a>, , <a class="ref" href="setwop_2.html#T9">SETWOP_2:9</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="setwop_2.html#D2">SETWOP_2:def 2</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#D8">FVSUM_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">A</font>,<font color="Maroon">i</font></span>)</span> <a href="fvsum_1.html#K13">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#D10">FVSUM_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
<b>by </b><i>, , , , </i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E19:43"/>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">B</font></span>)</span> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="matrix_3.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="matrix_3.html#K4" target="_self">*</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="matrix_1.html#T21">MATRIX_1:21</a></i>;<br/>


</div>
