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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">M1</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">M2</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="E1:50"/><i><font color="Green">E36</font></i>: 
( <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="E3:50"/><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="E4:50"/><i><font color="Green">E38</font></i>: 
(  <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> <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">M2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>
<a NAME="E5:50"/><i><font color="Green">E39</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M4</font> <a href="matrix_1.html#K5">@</a> </span>)</span> <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_1.html#K5">@</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>
<a NAME="E6:50"/><i><font color="Green">E40</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">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="hidden.html#R1">=</a> <font color="Maroon">n</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">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="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>
<i><font color="Green">E41</font></i>: <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> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M4</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<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></span>)</span> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M4</font>
<b>by </b><i>, , , <a class="ref" href="matrix_4.html#T62">MATRIX_4:62</a></i>
<br/>.= 
<font color="Maroon">M1</font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">M1</font>
<b>by </b><i>, , , , <a class="ref" href="matrix_4.html#T63">MATRIX_4:63</a></i>
;<br/>
<b>take </b>
<font color="Maroon">M4</font>
; <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="E8:50"/>
( <font color="Maroon">M4</font> is <a href="matrix_6.html#V1">invertible</a> &amp; <font color="Maroon">M1</font> <a href="matrix_6.html#K2">+</a> <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> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_6.html#K2">+</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="matrix_6.html#K4">*</a> <font color="Maroon">M4</font> )
 <b>by </b><i>, </i>;<br/>


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