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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">M</font> be   <a href="matrix_1.html#NM4">Matrix</a> of <font color="Maroon">K</font>;<br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1"><b>suppose </b></a><a NAME="E1:2_1_1"/><i><font color="Green">E28</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:2_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K2">-</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="matrix_3.html#D2">MATRIX_3:def 2</a></i>;<br/><a NAME="E3:2_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K2">-</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K2">-</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="matrix_3.html#D2">MATRIX_3:def 2</a></i>;<br/><b>hence </b><a NAME="E4:2_1_1"/>
 <a href="matrix_3.html#K2">-</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K2">-</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font>
 <b>by </b><i>, <a class="ref" href="matrix_1.html#T37">MATRIX_1:37</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_2"><b>suppose </b></a><a NAME="E1:2_1_2"/><i><font color="Green">E29</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:2_1_2"/><i><font color="Green">E30</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K2">-</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K2">-</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#D2">MATRIX_3:def 2</a></i>;<br/><a NAME="E3:2_1_2"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K2">-</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K2">-</a> <font color="Maroon">M</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_3.html#K2">-</a> <font color="Maroon">M</font></span>)</span> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K2">-</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K2">-</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K2">-</a> <font color="Maroon">M</font></span>)</span> )
 <b>by </b><i><a class="ref" href="matrix_3.html#D2">MATRIX_3:def 2</a></i>;<br/><a NAME="E4:2_1_2"/><b>then </b><i><font color="Green">E31</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K2">-</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K2">-</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K2">-</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K2">-</a> <font color="Maroon">M</font></span>)</span></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#D2">MATRIX_3:def 2</a></i>;<br/><a NAME="E5:2_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">M</font> is   <a href="matrix_1.html#NM6">Matrix</a> of  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font>, <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font>,<font color="Maroon">K</font>
 <b>by </b><i>, <a class="ref" href="matrix_1.html#T20">MATRIX_1:20</a></i>;<br/><a NAME="E6:2_1_2"/>
 <a href="matrix_3.html#K2">-</a> <font color="Maroon">M</font> is   <a href="matrix_1.html#NM6">Matrix</a> of  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font>, <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font>,<font color="Maroon">K</font>
 <b>by </b><i>, , <a class="ref" href="matrix_1.html#T20">MATRIX_1:20</a></i>;<br/><a NAME="E7:2_1_2"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K2">-</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M</font>
 <b>by </b><i>, <a class="ref" href="matrix_1.html#T27">MATRIX_1:27</a></i>;<br/><a NAME="E8:2_1_2"/>
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M</font> holds <br/><font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K2">-</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K2">-</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_2_1"><b>proof </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="E1:2_1_2_1"/><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M</font>
 ;<br/>

<a NAME="E2:2_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="matrix_3.html#K2">-</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>  <a href="rlvect_1.html#K5">-</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#D2">MATRIX_3:def 2</a></i>;<br/>
<a NAME="E3:2_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="matrix_3.html#K2">-</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K2">-</a> <font color="Maroon">M</font></span>)</span></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>  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="matrix_3.html#D2">MATRIX_3:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:2_1_2_1"/>
<font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K2">-</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K2">-</a> <font color="Maroon">M</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><a class="ref" href="rlvect_1.html#T30">RLVECT_1:30</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E9:2_1_2"/>
 <a href="matrix_3.html#K2">-</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K2">-</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font>
 <b>by </b><i>, <a class="ref" href="matrix_1.html#T21">MATRIX_1:21</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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