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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">K</font> being  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/> for <font color="Olive" title="b2">A</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="Olive" title="b1">K</font>  st <font color="Olive" title="b2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="matrix_1.html#K11" title="MATRIX_1:func.11">0.</a> <font color="Olive" title="b1">K</font>,<font color="Maroon" title="c1">n</font> holds <br/> ex <font color="Olive" title="b3">B</font>, <font color="Olive" title="b4">C</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="Olive" title="b1">K</font> st <br/>( <font color="Olive" title="b3">B</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> &amp; <font color="Olive" title="b4">C</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">B</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b2">A</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b4">C</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Olive" title="b1">K</font> &amp; ( for <font color="Olive" title="b5">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">B</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b2">A</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b4">C</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b5">i</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">K</font> ) &amp; ( for <font color="Olive" title="b5">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">j</font> &amp; <font color="Olive" title="b5">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">B</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b2">A</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b4">C</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">K</font> ) )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</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 <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="matrix_1.html#K11" title="MATRIX_1:func.11">0.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font> holds <br/> ex <font color="Olive" title="b2">B</font>, <font color="Olive" title="b3">C</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="b2">B</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> &amp; <font color="Olive" title="b3">C</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">B</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b3">C</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font> &amp; ( for <font color="Olive" title="b4">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">B</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b3">C</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">i</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> ) &amp; ( for <font color="Olive" title="b4">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">j</font> &amp; <font color="Olive" title="b4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">B</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b3">C</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> ) )</span><br/><b>let </b><font color="Maroon" title="c3">A</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"> ( <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="matrix_1.html#K11" title="MATRIX_1:func.11">0.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font> implies  ex <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</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">B</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> &amp; <font color="Olive" title="b2">C</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">B</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b2">C</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font> &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">B</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b2">C</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> ) &amp; ( for <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">B</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b2">C</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> ) ) )</span><br/>





<b>assume </b><a NAME="E1:60"/><i><font color="Green" title="E50">A0</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="matrix_1.html#K11" title="MATRIX_1:func.11">0.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</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">B</font>, <font color="Olive" title="b2">C</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">B</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> &amp; <font color="Olive" title="b2">C</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">B</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b2">C</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font> &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">B</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b2">C</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> ) &amp; ( for <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">B</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b2">C</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> ) )</span><br/>

<b>consider </b><font color="Maroon" title="c4">i0</font>, <font color="Maroon" title="c5">j0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:60"/><i><font color="Green" title="E51">A4</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i0</font> &amp; <font color="Maroon" title="c4">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j0</font> &amp; <font color="Maroon" title="c5">j0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i0</font>,<font color="Maroon" title="c5">j0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> )
 <b>by </b><i><a class="ref" href="matrix14.html#T52" target="_self" title="MATRIX14:th.52">AA4190</a>, <a class="txt" href="matrix14.html#E1:60"><i><font color="Green" title="E50">A0</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">A2</font> = <span class="p1">(<span class="default"><a href="matrix14.html#K5" title="MATRIX14:func.5">SwapDiagonal</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">i0</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">A</font>;<br/>
<a NAME="E4:60"/><i><font color="Green" title="E52">B1</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E3:60"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<i><font color="Green" title="E53">B3</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix14.html#K5" title="MATRIX14:func.5">SwapDiagonal</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">i0</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K5" title="MATRIX14:func.5">SwapDiagonal</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">j0</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K5" title="MATRIX14:func.5">SwapDiagonal</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">i0</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c5">j0</font>
<b>by </b><i><a class="txt" href="matrix14.html#E3:60"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="matrix14.html#T51" target="_self" title="MATRIX14:th.51">AA4170</a>, <a class="txt" href="matrix14.html#E4:60"><i><font color="Green" title="E52">B1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i0</font>,<font color="Maroon" title="c5">j0</font>
<b>by </b><i><a class="txt" href="matrix14.html#E3:60"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="matrix14.html#T48" target="_self" title="MATRIX14:th.48">AA4100</a></i>
;<br/>
<b>set </b><font color="Maroon" title="c7">A3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K5" title="MATRIX14:func.5">SwapDiagonal</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">i0</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><a href="matrix14.html#K5" title="MATRIX14:func.5">SwapDiagonal</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">j0</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c8">P</font>, <font color="Maroon" title="c9">Q</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><br/><a NAME="E7:60"/><i><font color="Green" title="E54">A13</font></i>: 
( <font color="Maroon" title="c8">P</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> &amp; <font color="Maroon" title="c9">Q</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">P</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"><span class="p5">(<span class="default"><a href="matrix14.html#K5" title="MATRIX14:func.5">SwapDiagonal</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">i0</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K5" title="MATRIX14:func.5">SwapDiagonal</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">j0</font></span>)</span></span>)</span></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c9">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">P</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"><span class="p5">(<span class="default"><a href="matrix14.html#K5" title="MATRIX14:func.5">SwapDiagonal</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">i0</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K5" title="MATRIX14:func.5">SwapDiagonal</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">j0</font></span>)</span></span>)</span></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c9">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> ) &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">P</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"><span class="p5">(<span class="default"><a href="matrix14.html#K5" title="MATRIX14:func.5">SwapDiagonal</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">i0</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K5" title="MATRIX14:func.5">SwapDiagonal</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">j0</font></span>)</span></span>)</span></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c9">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> ) )
 <b>by </b><i><a class="txt" href="matrix14.html#E3:60"><i><font color="Green" title="E51">A4</font></i></a>, <a class="txt" href="matrix14.html#E5:60"><i><font color="Green" title="E53">B3</font></i></a>, <a class="ref" href="matrix14.html#T41" target="_self" title="MATRIX14:th.41">AA3040</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">B0</font> = <font color="Maroon" title="c8">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><a href="matrix14.html#K5" title="MATRIX14:func.5">SwapDiagonal</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">i0</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c11">C0</font> = <span class="p1">(<span class="default"><a href="matrix14.html#K5" title="MATRIX14:func.5">SwapDiagonal</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">j0</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c9">Q</font>;<br/>
<a NAME="E8:60"/><i><font color="Green" title="E55">A15a</font></i>: 
 <a href="matrix14.html#K5" title="MATRIX14:func.5">SwapDiagonal</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">i0</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>
 
<b>by </b><i><a class="txt" href="matrix14.html#E3:60"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="matrix14.html#T49" target="_self" title="MATRIX14:th.49">AA4150</a></i>;<br/>
<a NAME="E9:60"/><i><font color="Green" title="E56">A15</font></i>: 
<font color="Maroon" title="c8">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><a href="matrix14.html#K5" title="MATRIX14:func.5">SwapDiagonal</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">i0</font></span>)</span> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>
 
<b>by </b><i><a class="txt" href="matrix14.html#E7:60"><i><font color="Green" title="E54">A13</font></i></a>, <a class="ref" href="matrix_6.html#T37" title="MATRIX_6:th.37">MATRIX_6:37</a>, <a class="txt" href="matrix14.html#E8:60"><i><font color="Green" title="E55">A15a</font></i></a></i>;<br/>
<a NAME="E10:60"/><i><font color="Green" title="E57">A15b</font></i>: 
 <a href="matrix14.html#K5" title="MATRIX14:func.5">SwapDiagonal</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">j0</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>
 
<b>by </b><i><a class="txt" href="matrix14.html#E3:60"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="matrix14.html#T49" target="_self" title="MATRIX14:th.49">AA4150</a></i>;<br/>
<a NAME="E11:60"/><i><font color="Green" title="E58">A17</font></i>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K5" title="MATRIX14:func.5">SwapDiagonal</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">j0</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c9">Q</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>
 
<b>by </b><i><a class="txt" href="matrix14.html#E7:60"><i><font color="Green" title="E54">A13</font></i></a>, <a class="ref" href="matrix_6.html#T37" title="MATRIX_6:th.37">MATRIX_6:37</a>, <a class="txt" href="matrix14.html#E10:60"><i><font color="Green" title="E57">A15b</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K5" title="MATRIX14:func.5">SwapDiagonal</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">i0</font></span>)</span></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">A</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"><a href="matrix14.html#K5" title="MATRIX14:func.5">SwapDiagonal</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">j0</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c9">Q</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">P</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"><a href="matrix14.html#K5" title="MATRIX14:func.5">SwapDiagonal</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">i0</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">A</font></span>)</span></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"><a href="matrix14.html#K5" title="MATRIX14:func.5">SwapDiagonal</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">j0</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c9">Q</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix14.html#T17" target="_self" title="MATRIX14:th.17">AA41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">P</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"><a href="matrix14.html#K5" title="MATRIX14:func.5">SwapDiagonal</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">i0</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K5" title="MATRIX14:func.5">SwapDiagonal</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">j0</font></span>)</span></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c9">Q</font>
<b>by </b><i><a class="ref" href="matrix14.html#T17" target="_self" title="MATRIX14:th.17">AA41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">P</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"><a href="matrix14.html#K5" title="MATRIX14:func.5">SwapDiagonal</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">i0</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K5" title="MATRIX14:func.5">SwapDiagonal</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c5">j0</font></span>)</span></span>)</span></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c9">Q</font>
<b>by </b><i><a class="ref" href="matrix14.html#T17" target="_self" title="MATRIX14:th.17">AA41</a></i>
;<br/>
<b>hence </b><a NAME="E13:60"/>
 ex <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</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">B</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> &amp; <font color="Olive" title="b2">C</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">B</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b2">C</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font> &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">B</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b2">C</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> ) &amp; ( for <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">B</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b2">C</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> ) )
 <b>by </b><i><a class="txt" href="matrix14.html#E9:60"><i><font color="Green" title="E56">A15</font></i></a>, <a class="txt" href="matrix14.html#E11:60"><i><font color="Green" title="E58">A17</font></i></a>, <a class="txt" href="matrix14.html#E7:60"><i><font color="Green" title="E54">A13</font></i></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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