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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><a NAME="E1:64"/><i><font color="Green">E52</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font> ) )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:64"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="matrix_9.html#E1:64"><i><font color="Green">E52</font></i></a>, <a class="ref" href="matrix_9.html#T49" target="_self">Th49</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font>;<br/>
<a NAME="E4:64"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E3:64"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E5:64"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="matrix_3.html#D7">MATRIX_3:def 7</a></i>;<br/>
<a NAME="E6:64"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E3:64"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:64"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E4:64"><i><font color="Green">E54</font></i></a>, <a class="ref" href="fvsum_1.html#T107">FVSUM_1:107</a></i>;<br/>
<i><font color="Green">E56</font></i>: <span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> = 
 <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>2</sub></font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="matrix_3.html#D8">MATRIX_3:def 8</a></i>
<br/>.= 
 <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default"><a href="group_4.html#K3">Product</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="group_4.html#D3">GROUP_4:def 3</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_2"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="matrix_2.html#V3">even</a> or  not <font color="Maroon">c<sub>3</sub></font> is <a href="matrix_2.html#V3">even</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_1"><b>suppose </b></a><a NAME="E1:64_2_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="matrix_2.html#V3">even</a>
 ;<br/><div class="add"><b>then </b> <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default"><a href="group_4.html#K3">Product</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font> = 
 <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_2.html#D16">MATRIX_2:def 16</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="matrix_9.html#E7:64"><i><font color="Green">E55</font></i></a></i>
;<br/><b>hence </b><a NAME="E3:64_2_1"/>
<span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="matrix_9.html#E8:64"><i><font color="Green">E56</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_2"><b>suppose </b></a><a NAME="E1:64_2_2"/>
not <font color="Maroon">c<sub>3</sub></font> is <a href="matrix_2.html#V3">even</a>
 ;<br/><div class="add"><b>then </b> <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default"><a href="group_4.html#K3">Product</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font> = 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="group_4.html#K3">Product</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_2.html#D16">MATRIX_2:def 16</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="matrix_9.html#E7:64"><i><font color="Green">E55</font></i></a>, <a class="ref" href="rlvect_1.html#T25">RLVECT_1:25</a></i>
;<br/><b>hence </b><a NAME="E3:64_2_2"/>
<span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="matrix_9.html#E8:64"><i><font color="Green">E56</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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