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<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="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">M</font> be   <a href="matrix_1.html#NM4">Matrix</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>







<b>assume </b><a NAME="E1:20"/><i><font color="Green">E31</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> )
 ;<br/>

<a NAME="E2:20"/><i><font color="Green">E54</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="fvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:20"/><i><font color="Green">E55</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">M</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E4:20"/><i><font color="Green">E56</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="matrix_3.html#D5">MATRIX_3:def 5</a></i>;<br/>
<a NAME="E5:20"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="fvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, , </i>;<br/>
<a NAME="E6:20"/><i><font color="Green">E58</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_3.html#D5">MATRIX_3:def 5</a></i>;<br/>
<a NAME="E7:20"/>
for <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="fvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:20_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:20_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E3:20_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M</font>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E4:20_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_3.html#D5">MATRIX_3:def 5</a></i>;<br/>
<a NAME="E5:20_1"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a>, , </i>;<br/>
<b>hence </b><a NAME="E6:20_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="fvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="fvsum_1.html#T62">FVSUM_1:62</a>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:20"/>
 <a href="matrix_1.html#K8">Line</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="fvsum_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a>, , , </i>;<br/>


</div>
