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<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">K</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">M2</font> be   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">M1</font> be   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">K</font>;<br/>







<b>assume </b><a NAME="E1:53"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">M2</font> is <a href="matrix_6.html#V1">invertible</a> &amp; <font color="Maroon">M1</font> <a href="matrix_8.html#R2" target="_self">is_congruent_Matrix_of</a> <font color="Maroon">M2</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<b>then consider </b><font color="Maroon">M4</font> being   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">K</font><b> such that </b><br/><a NAME="E3:53"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">M4</font> is <a href="matrix_6.html#V1">invertible</a> &amp; <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M4</font> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M4</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:53"/><i><font color="Green">E38</font></i>: 
(  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M4</font> <a href="matrix_6.html#K5">~</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M4</font> <a href="matrix_6.html#K5">~</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M4</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i>, <a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>
<a NAME="E5:53"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">M4</font> <a href="matrix_6.html#K5">~</a>  <a href="matrix_6.html#R2">is_reverse_of</a> <font color="Maroon">M4</font>
 
<b>by </b><i>, <a class="ref" href="matrix_6.html#D4">MATRIX_6:def 4</a></i>;<br/>
<a NAME="E6:53"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">M4</font> <a href="matrix_6.html#K5">~</a> </span>)</span> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M4</font> <a href="matrix_6.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">M4</font> <a href="matrix_6.html#K5">~</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">M4</font> <a href="matrix_6.html#K5">~</a> </span>)</span> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M4</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_6.html#D2">MATRIX_6:def 2</a></i>;<br/>
<a NAME="E7:53"/><b>then </b><i><font color="Green">E40</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M4</font> <a href="matrix_6.html#K5">~</a> </span>)</span> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M4</font></span>)</span> <a href="matrix_1.html#K5">@</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M4</font> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="matrix_6.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M4</font> <a href="matrix_6.html#K5">~</a> </span>)</span> <a href="matrix_1.html#K5">@</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">M4</font> <a href="matrix_6.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">M4</font> <a href="matrix_6.html#K5">~</a> </span>)</span></span>)</span> <a href="matrix_1.html#K5">@</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M4</font> <a href="matrix_6.html#K5">~</a> </span>)</span> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="matrix_6.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">M4</font> <a href="matrix_1.html#K5">@</a> </span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M4</font> <a href="matrix_6.html#K5">~</a> </span>)</span> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M4</font></span>)</span> <a href="matrix_1.html#K5">@</a>  <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font> )
 
<b>by </b><i>, <a class="ref" href="matrix_3.html#T24">MATRIX_3:24</a>, <a class="ref" href="matrix_6.html#T10">MATRIX_6:10</a></i>;<br/>
<b>set </b><font color="Maroon">M5</font> = <font color="Maroon">M4</font> <a href="matrix_1.html#K5">@</a> ;<br/>
<b>set </b><font color="Maroon">M6</font> = <span class="p1">(<span class="default"><font color="Maroon">M4</font> <a href="matrix_6.html#K5">~</a> </span>)</span> <a href="matrix_1.html#K5">@</a> ;<br/>
<a NAME="E8:53"/>
( <span class="p1">(<span class="default"><font color="Maroon">M4</font> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="matrix_6.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M4</font> <a href="matrix_6.html#K5">~</a> </span>)</span> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M4</font> <a href="matrix_6.html#K5">~</a> </span>)</span> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="matrix_6.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">M4</font> <a href="matrix_1.html#K5">@</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">M4</font> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="matrix_6.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M4</font> <a href="matrix_6.html#K5">~</a> </span>)</span> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K12">1.</a> <font color="Maroon">K</font>,<font color="Maroon">n</font> )
 
<b>by </b><i>, <a class="ref" href="matrix_6.html#D2">MATRIX_6:def 2</a>, </i>;<br/>
<a NAME="E9:53"/><b>then </b>
<font color="Maroon">M4</font> <a href="matrix_1.html#K5">@</a>  <a href="matrix_6.html#R2">is_reverse_of</a> <span class="p1">(<span class="default"><font color="Maroon">M4</font> <a href="matrix_6.html#K5">~</a> </span>)</span> <a href="matrix_1.html#K5">@</a> 
 
<b>by </b><i><a class="ref" href="matrix_6.html#D2">MATRIX_6:def 2</a></i>;<br/>
<a NAME="E10:53"/><b>then </b>
<font color="Maroon">M4</font> <a href="matrix_1.html#K5">@</a>  is <a href="matrix_6.html#V1">invertible</a>
 
<b>by </b><i><a class="ref" href="matrix_6.html#D3">MATRIX_6:def 3</a></i>;<br/>
<a NAME="E11:53"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">M4</font> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M2</font> is <a href="matrix_6.html#V1">invertible</a>
 
<b>by </b><i>, <a class="ref" href="matrix_6.html#T37">MATRIX_6:37</a></i>;<br/>
<b>hence </b><a NAME="E12:53"/>
<font color="Maroon">M1</font> is <a href="matrix_6.html#V1">invertible</a>
 <b>by </b><i>, <a class="ref" href="matrix_6.html#T37">MATRIX_6:37</a></i>;<br/>


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