<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">M2</font>, <font color="Olive" title="b3">M1</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">M2</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  &amp; <font color="Olive" title="b3">M1</font> <a href="matrix_8.html#R1" title="MATRIX_8:pred.1">is_similar_to</a> <font color="Olive" title="b2">M2</font> holds <br/><font color="Olive" title="b3">M1</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  <a href="matrix_8.html#R1" title="MATRIX_8:pred.1">is_similar_to</a> <font color="Olive" title="b2">M2</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">M2</font>, <font color="Olive" title="b2">M1</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">M2</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  &amp; <font color="Olive" title="b2">M1</font> <a href="matrix_8.html#R1" title="MATRIX_8:pred.1">is_similar_to</a> <font color="Olive" title="b1">M2</font> holds <br/><font color="Olive" title="b2">M1</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  <a href="matrix_8.html#R1" title="MATRIX_8:pred.1">is_similar_to</a> <font color="Olive" title="b1">M2</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">M2</font>, <font color="Maroon" title="c4">M1</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">M2</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  &amp; <font color="Maroon" title="c4">M1</font> <a href="matrix_8.html#R1" title="MATRIX_8:pred.1">is_similar_to</a> <font color="Maroon" title="c3">M2</font> implies <font color="Maroon" title="c4">M1</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  <a href="matrix_8.html#R1" title="MATRIX_8:pred.1">is_similar_to</a> <font color="Maroon" title="c3">M2</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:46"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
<font color="Maroon" title="c3">M2</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> 
 <span class="kw">and </span><br/><a NAME="E2:46"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
<font color="Maroon" title="c4">M1</font> <a href="matrix_8.html#R1" title="MATRIX_8:pred.1">is_similar_to</a> <font color="Maroon" title="c3">M2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">M1</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  <a href="matrix_8.html#R1" title="MATRIX_8:pred.1">is_similar_to</a> <font color="Maroon" title="c3">M2</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">M4</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><span class="kw"> such that </span><br/><a NAME="E4:46"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
<font color="Maroon" title="c5">M4</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> 
 <span class="kw">and </span><br/><a NAME="E5:46"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
<font color="Maroon" title="c4">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="c5">M4</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">M2</font></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">M4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="matrix_8.html#D5" target="_self" title="MATRIX_8:def.5">Def5</a></span>;<br/>
<a NAME="E6:46"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
( <font color="Maroon" title="c5">M4</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; <span class="p1">(<span class="default"><font color="Maroon" title="c5">M4</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>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">M4</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="matrix_6.html#T16" title="MATRIX_6:th.16">MATRIX_6:16</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">M4</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="matrix_8.html#D5" target="_self" title="MATRIX_8:def.5">MATRIX_8:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">M4</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  &amp; <font color="Maroon" title="c4">M1</font> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">M4</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">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> <font color="Maroon" title="c5">M4</font> )</span><br/>

<a NAME="E7:46"/><span class="lab"><font color="Green" title="E15">A6</font></span>: 
(  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M4</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> <font color="Maroon" title="c5">M4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/>
<a NAME="E8:46"/><span class="lab"><font color="Green" title="E16">A7</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">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="c3">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/>
<a NAME="E9:46"/>
( <font color="Maroon" title="c3">M2</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">M4</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="c5">M4</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">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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M2</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">M4</font></span>)</span> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="matrix_6.html#T37" title="MATRIX_6:th.37">MATRIX_6:37</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">M4</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">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> <font color="Maroon" title="c5">M4</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">M4</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">M2</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">M4</font></span>)</span></span>)</span> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="matrix_6.html#T37" title="MATRIX_6:th.37">MATRIX_6:37</a></span>
<br/>.= 
<font color="Maroon" title="c4">M1</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E7:46"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="txt" href="#E8:46"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="ref" href="matrix_3.html#T35" title="MATRIX_3:th.35">MATRIX_3:35</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E11:46"/>
( <font color="Maroon" title="c5">M4</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  &amp; <font color="Maroon" title="c4">M1</font> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">M4</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">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> <font color="Maroon" title="c5">M4</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E12">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
