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<b>let </b><font color="Maroon" title="c5">M1</font>, <font color="Maroon" title="c6">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">M</font> being  <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> st <br/>( <font color="Olive" title="b1">M</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> &amp; <font color="Maroon" title="c5">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">M</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="c6">M2</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b1">M</font> ) implies  ex <font color="Olive" title="b1">M</font> being  <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> st <br/>( <font color="Olive" title="b1">M</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> &amp; <font color="Maroon" title="c6">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">M</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="c5">M1</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b1">M</font> ) )</span><br/>

<b>given </b><font color="Maroon" title="c7">M</font> being   <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><b> such that </b><a NAME="E2:36_1_2"/><i><font color="Green" title="E9">A2</font></i>: 
( <font color="Maroon" title="c7">M</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> &amp; <font color="Maroon" title="c5">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">M</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="c6">M2</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c7">M</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">M</font> being  <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> st <br/>( <font color="Olive" title="b1">M</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> &amp; <font color="Maroon" title="c6">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">M</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="c5">M1</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b1">M</font> )</span><br/>

<a NAME="E3:36_1_2"/><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c7">M</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="c7">M</font>
 
<b>by </b><i><a class="txt" href="matrix_8.html#E2:36_1_2"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="matrix_6.html#D4" title="MATRIX_6:def.4">MATRIX_6:def 4</a></i>;<br/>
<a NAME="E4:36_1_2"/><i><font color="Green" title="E11">A4</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">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="c6">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="c7">M</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="c7">M</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="E5:36_1_2"/><i><font color="Green" title="E12">A5</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">M</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="c7">M</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="E6:36_1_2"/><i><font color="Green" title="E13">A6</font></i>: 
(  <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="c7">M</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="c6">M2</font></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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">M</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="c6">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/>
<i><font color="Green" title="E14">A7</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">M</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span> <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="c5">M1</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">M</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">M</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">M</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="c6">M2</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c7">M</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="c7">M</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span>
<b>by </b><i><a class="txt" href="matrix_8.html#E2:36_1_2"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="matrix_6.html#T16" title="MATRIX_6:th.16">MATRIX_6:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">M</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">M</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="c6">M2</font></span>)</span></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c7">M</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">M</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span>
<b>by </b><i><a class="txt" href="matrix_8.html#E4:36_1_2"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="matrix_8.html#E6:36_1_2"><i><font color="Green" title="E13">A6</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"><span class="p3">(<span class="default"><font color="Maroon" title="c7">M</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">M</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="c6">M2</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c7">M</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">M</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span>
<b>by </b><i><a class="txt" href="matrix_8.html#E4:36_1_2"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="matrix_8.html#E5:36_1_2"><i><font color="Green" title="E12">A5</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"><span class="p3">(<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> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c7">M</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">M</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span>
<b>by </b><i><a class="txt" href="matrix_8.html#E3:36_1_2"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="matrix_6.html#D2" title="MATRIX_6:def.2">MATRIX_6:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c7">M</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">M</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#T20" title="MATRIX_3:th.20">MATRIX_3:20</a></i>
<br/>.= 
<font color="Maroon" title="c6">M2</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">M</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">M</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix_8.html#E4:36_1_2"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="matrix_8.html#E5:36_1_2"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="matrix_3.html#T35" title="MATRIX_3:th.35">MATRIX_3:35</a></i>
<br/>.= 
<font color="Maroon" title="c6">M2</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<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>
<b>by </b><i><a class="txt" href="matrix_8.html#E3:36_1_2"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="matrix_6.html#D2" title="MATRIX_6:def.2">MATRIX_6:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c6">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/>
<b>take </b>
<font color="Maroon" title="c7">M</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">M</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> &amp; <font color="Maroon" title="c6">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">M</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span> <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="c5">M1</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">M</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span> )</span><br/>

<b>thus </b><a NAME="E8:36_1_2"/>
( <font color="Maroon" title="c7">M</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> &amp; <font color="Maroon" title="c6">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">M</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span> <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="c5">M1</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">M</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="matrix_8.html#E2:36_1_2"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="matrix_8.html#E7:36_1_2"><i><font color="Green" title="E14">A7</font></i></a>, <a class="ref" href="matrix_6.html#T16" title="MATRIX_6:th.16">MATRIX_6:16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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