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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c2">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c3">M1</font>, <font color="Maroon" title="c4">M2</font> be   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>;<br/>







<b>assume </b><a NAME="E1:46"/><i><font color="Green" title="E19">A1</font></i>: 
( <font color="Maroon" title="c3">M1</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> &amp; <font color="Maroon" title="c4">M2</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> )
 ;<br/>

<a NAME="E2:46"/><i><font color="Green" title="E20">A2</font></i>: 
(  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<a NAME="E3:46"/><i><font color="Green" title="E21">A3</font></i>: 
( <font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  is   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c4">M2</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  is   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<a NAME="E4:46"/><i><font color="Green" title="E22">A4</font></i>: 
(  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">M2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">M2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<a NAME="E5:46"/><i><font color="Green" title="E23">A5</font></i>: 
( <font color="Maroon" title="c4">M2</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  <a href="matrix_6.html#R2" title="MATRIX_6:pred.2">is_reverse_of</a> <font color="Maroon" title="c4">M2</font> &amp; <font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  <a href="matrix_6.html#R2" title="MATRIX_6:pred.2">is_reverse_of</a> <font color="Maroon" title="c3">M1</font> )
 
<b>by </b><i><a class="txt" href="matrix_6.html#E1:46"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="matrix_6.html#D4" target="_self" title="MATRIX_6:def.4">Def4</a></i>;<br/>
<i><font color="Green" title="E24">A6</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">M2</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">M1</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">M2</font>
<b>by </b><i><a class="txt" href="matrix_6.html#E2:46"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="matrix_6.html#E4:46"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="matrix_3.html#T35" title="MATRIX_3:th.35">MATRIX_3:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">M1</font></span>)</span></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">M2</font>
<b>by </b><i><a class="txt" href="matrix_6.html#E2:46"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="matrix_6.html#E3:46"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="matrix_3.html#T35" title="MATRIX_3:th.35">MATRIX_3:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">M2</font>
<b>by </b><i><a class="txt" href="matrix_6.html#E5:46"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="matrix_6.html#D2" target="_self" title="MATRIX_6:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">M2</font>
<b>by </b><i><a class="ref" href="matrix_3.html#T21" title="MATRIX_3:th.21">MATRIX_3:21</a></i>
<br/>.= 
 <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="matrix_6.html#E5:46"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="matrix_6.html#D2" target="_self" title="MATRIX_6:def.2">Def2</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">M2</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">M2</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span>
<b>by </b><i><a class="txt" href="matrix_6.html#E3:46"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="matrix_6.html#E4:46"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="matrix_3.html#T35" title="MATRIX_3:th.35">MATRIX_3:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span></span>)</span></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span>
<b>by </b><i><a class="txt" href="matrix_6.html#E2:46"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="matrix_6.html#E3:46"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="matrix_3.html#T35" title="MATRIX_3:th.35">MATRIX_3:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span>
<b>by </b><i><a class="txt" href="matrix_6.html#E5:46"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="matrix_6.html#D2" target="_self" title="MATRIX_6:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#T21" title="MATRIX_3:th.21">MATRIX_3:21</a></i>
<br/>.= 
 <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="matrix_6.html#E5:46"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="matrix_6.html#D2" target="_self" title="MATRIX_6:def.2">Def2</a></i>
;<br/>
<a NAME="E8:46"/><b>then </b><i><font color="Green" title="E25">A7</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span> <a href="matrix_6.html#R2" title="MATRIX_6:pred.2">is_reverse_of</a> <font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">M2</font>
 
<b>by </b><i><a class="txt" href="matrix_6.html#E6:46"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="matrix_6.html#D2" target="_self" title="MATRIX_6:def.2">Def2</a></i>;<br/>
<a NAME="E9:46"/><b>then </b>
<font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">M2</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>
 
<b>by </b><i><a class="ref" href="matrix_6.html#D3" target="_self" title="MATRIX_6:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E10:46"/>
( <font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">M2</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">M2</font></span>)</span> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="matrix_6.html#D4" target="_self" title="MATRIX_6:def.4">Def4</a>, <a class="txt" href="matrix_6.html#E8:46"><i><font color="Green" title="E25">A7</font></i></a></i>;<br/>


</div>
