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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>, <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="E1:48"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="matrix_8.html#R2" target="_self">is_congruent_Matrix_of</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <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><b> such that </b><br/><a NAME="E3:48"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> is <a href="matrix_6.html#V1">Reversible</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="matrix_8.html#E1:48"><i><font color="Green">E9</font></i></a>, <a class="ref" href="matrix_8.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E4:48"/><i><font color="Green">E11</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>
<a NAME="E5:48"/><i><font color="Green">E12</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="matrix_6.html#K5">~</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="matrix_6.html#K5">~</a> </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_1.html#T25">MATRIX_1:25</a></i>;<br/>
<a NAME="E6:48"/><i><font color="Green">E13</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="matrix_8.html#E1:48"><i><font color="Green">E9</font></i></a>, <a class="txt" href="matrix_8.html#E4:48"><i><font color="Green">E11</font></i></a>, <a class="ref" href="matrix_2.html#T12">MATRIX_2:12</a></i>;<br/>
<a NAME="E7:48"/><i><font color="Green">E14</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>
<a NAME="E8:48"/><b>then </b><i><font color="Green">E15</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">c<sub>5</sub></font> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></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">c<sub>5</sub></font> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="matrix_6.html#K4">*</a> <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="txt" href="matrix_8.html#E6:48"><i><font color="Green">E13</font></i></a>, <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>;<br/>
<a NAME="E9:48"/><b>then </b><i><font color="Green">E16</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font> is   <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>
 
<b>by </b><i><a class="txt" href="matrix_8.html#E1:48"><i><font color="Green">E9</font></i></a>, <a class="ref" href="matrix_1.html#T20">MATRIX_1:20</a></i>;<br/>
<a NAME="E10:48"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="matrix_6.html#K5">~</a>  <a href="matrix_6.html#R2">is_reverse_of</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="matrix_8.html#E3:48"><i><font color="Green">E10</font></i></a>, <a class="ref" href="matrix_6.html#D4">MATRIX_6:def 4</a></i>;<br/>
<a NAME="E11:48"/><i><font color="Green">E18</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">c<sub>5</sub></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> <font color="Maroon">c<sub>1</sub></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">c<sub>5</sub></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> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="matrix_8.html#E1:48"><i><font color="Green">E9</font></i></a>, <a class="txt" href="matrix_8.html#E5:48"><i><font color="Green">E12</font></i></a>, <a class="ref" href="matrix_2.html#T12">MATRIX_2:12</a></i>;<br/>
<i><font color="Green">E19</font></i>: <font color="Maroon">c<sub>3</sub></font> <a href="matrix_6.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="matrix_6.html#K5">~</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_6.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="matrix_6.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="matrix_6.html#K5">~</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix_8.html#E3:48"><i><font color="Green">E10</font></i></a>, <a class="txt" href="matrix_8.html#E4:48"><i><font color="Green">E11</font></i></a>, <a class="txt" href="matrix_8.html#E5:48"><i><font color="Green">E12</font></i></a>, <a class="txt" href="matrix_8.html#E8:48"><i><font color="Green">E15</font></i></a>, <a class="ref" href="matrix_3.html#T35">MATRIX_3:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_6.html#K4">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_8.html#E10:48"><i><font color="Green">E17</font></i></a>, <a class="ref" href="matrix_6.html#D2">MATRIX_6:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="matrix_8.html#E9:48"><i><font color="Green">E16</font></i></a>, <a class="ref" href="matrix_3.html#T21">MATRIX_3:21</a></i>
;<br/>
<i><font color="Green">E20</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></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> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_6.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="matrix_6.html#K5">~</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font> <a href="matrix_6.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="matrix_6.html#K5">~</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix_8.html#E5:48"><i><font color="Green">E12</font></i></a>, <a class="txt" href="matrix_8.html#E7:48"><i><font color="Green">E14</font></i></a>, <a class="txt" href="matrix_8.html#E11:48"><i><font color="Green">E18</font></i></a>, <a class="ref" href="matrix_3.html#T35">MATRIX_3:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></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="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K5">@</a> </span>)</span></span>)</span> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="matrix_8.html#E12:48"><i><font color="Green">E19</font></i></a>, <a class="txt" href="matrix_8.html#E6:48"><i><font color="Green">E13</font></i></a>, <a class="txt" href="matrix_8.html#E7:48"><i><font color="Green">E14</font></i></a>, <a class="txt" href="matrix_8.html#E11:48"><i><font color="Green">E18</font></i></a>, <a class="ref" href="matrix_3.html#T35">MATRIX_3:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="matrix_6.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="matrix_6.html#K5">~</a> </span>)</span></span>)</span> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="matrix_8.html#E1:48"><i><font color="Green">E9</font></i></a>, <a class="txt" href="matrix_8.html#E4:48"><i><font color="Green">E11</font></i></a>, <a class="txt" href="matrix_8.html#E5:48"><i><font color="Green">E12</font></i></a>, <a class="ref" href="matrix_3.html#T24">MATRIX_3:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K5">@</a> </span>)</span> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="matrix_8.html#E10:48"><i><font color="Green">E17</font></i></a>, <a class="ref" href="matrix_6.html#D2">MATRIX_6:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_6.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="matrix_8.html#E1:48"><i><font color="Green">E9</font></i></a>, <a class="ref" href="matrix_6.html#T10">MATRIX_6:10</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="matrix_3.html#T20">MATRIX_3:20</a></i>
;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font> <a href="matrix_6.html#K5">~</a> 
; <i><font color="Red">:: according to </font></i><a class="ref" href="matrix_8.html#D6" target="_self">MATRIX_8:def 6</a><br/>

<b>thus </b><a NAME="E14:48"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="matrix_6.html#K5">~</a>  is <a href="matrix_6.html#V1">Reversible</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></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> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_6.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="matrix_6.html#K5">~</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="matrix_8.html#E3:48"><i><font color="Green">E10</font></i></a>, <a class="ref" href="matrix_6.html#T16">MATRIX_6:16</a>, <a class="txt" href="matrix_8.html#E13:48"><i><font color="Green">E20</font></i></a></i>;<br/>


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