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

<span class="kw">consider </span><font color="Maroon" title="c6">N2</font> being   <a href="matrix_1.html#NM2" title="MATRIX_1:NM.2">Matrix</a> of 3,<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a><span class="kw"> such that </span><br/><a NAME="E2:40_1_1"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<font color="Maroon" title="c5">N</font> <a href="matrix_6.html#R2" title="MATRIX_6:pred.2">is_reverse_of</a> <font color="Maroon" title="c6">N2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_6.html#D3" onmouseover="rs('matrix_6/D3')" onmouseout="rh()">MATRIX_6:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">b</font> = 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/>
<a NAME="E4:40_1_1"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>
 
;<br/>
<a NAME="E5:40_1_1"/>
 not <font color="Maroon" title="c4">a</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E31">A3</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">N2</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c5">N</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a></span>)</span> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">N2</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c5">N</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T106" onmouseover="rs('xcmplx_1/T106')" onmouseout="rh()">XCMPLX_1:106</a></span>
<br/>.= 
<font color="Maroon" title="c6">N2</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c5">N</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="matrix15.html#T2" onmouseover="rs('matrix15/T2')" onmouseout="rh()">MATRIX15:2</a></span>
;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_1_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Maroon" title="c6">N2</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Maroon" title="c6">N2</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Maroon" title="c6">N2</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K3" title="MATRIX_1:func.3">1.</a> (<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>,3) )</span></a><div class="add"><a NAME="E1:40_1_1_2"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
(  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c6">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp;  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c5">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T23" onmouseover="rs('matrix_0/T23')" onmouseout="rh()">MATRIX_0:23</a></span>;<br/><a NAME="E2:40_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A5</font></span>: 
(  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c6">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">N</font> &amp;  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c5">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">N2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T23" onmouseover="rs('matrix_0/T23')" onmouseout="rh()">MATRIX_0:23</a></span>;<br/><a NAME="E3:40_1_1_2"/><span class="lab"><font color="Green" title="E34">A6</font></span>: 
(  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c6">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Maroon" title="c5">N</font></span>)</span> &amp;  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c5">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Maroon" title="c6">N2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_1_2"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="matrix_0.html#T23" onmouseover="rs('matrix_0/T23')" onmouseout="rh()">MATRIX_0:23</a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Maroon" title="c6">N2</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Maroon" title="c5">N</font></span>)</span> = 
<font color="Maroon" title="c7">b</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">N2</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix15.html#T1" onmouseover="rs('matrix15/T1')" onmouseout="rh()">MATRIX15:1</a></span>
<br/>.= 
<font color="Maroon" title="c7">b</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">N2</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_1_2"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="ref" href="matrixr1.html#T22" onmouseover="rs('matrixr1/T22')" onmouseout="rh()">MATRIXR1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">N2</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c5">N</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix15.html#T2" onmouseover="rs('matrix15/T2')" onmouseout="rh()">MATRIX15:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">N</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">N2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_1"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="matrix_6.html#D2" onmouseover="rs('matrix_6/D2')" onmouseout="rh()">MATRIX_6:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c4">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">N</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">N2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix15.html#T2" onmouseover="rs('matrix15/T2')" onmouseout="rh()">MATRIX15:2</a></span>
<br/>.= 
<font color="Maroon" title="c4">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">N</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Maroon" title="c6">N2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_1_2"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="ref" href="matrixr1.html#T22" onmouseover="rs('matrixr1/T22')" onmouseout="rh()">MATRIXR1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Maroon" title="c6">N2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_1_2"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="ref" href="matrix15.html#T1" onmouseover="rs('matrix15/T1')" onmouseout="rh()">MATRIX15:1</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Maroon" title="c6">N2</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K3" title="MATRIX_1:func.3">1.</a> (<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>,3)</span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Maroon" title="c6">N2</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Maroon" title="c5">N</font></span>)</span> = 
<font color="Maroon" title="c7">b</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">N2</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_1_2"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="ref" href="matrix15.html#T1" onmouseover="rs('matrix15/T1')" onmouseout="rh()">MATRIX15:1</a></span>
<br/>.= 
<font color="Maroon" title="c7">b</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">N2</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c5">N</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_1_2"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="ref" href="matrixr1.html#T22" onmouseover="rs('matrixr1/T22')" onmouseout="rh()">MATRIXR1:22</a></span>
<br/>.= 
<font color="Maroon" title="c6">N2</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c5">N</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40_1_1"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="matrix15.html#T2" onmouseover="rs('matrix15/T2')" onmouseout="rh()">MATRIX15:2</a></span>
<br/>.= 
 <a href="matrix_1.html#K3" title="MATRIX_1:func.3">1.</a> (<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>,3)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_1"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="matrix_6.html#D2" onmouseover="rs('matrix_6/D2')" onmouseout="rh()">MATRIX_6:def 2</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:40_1_1"/>
 for <font color="Olive">b<sub>1</sub></font> being   <a href="matrix_1.html#NM2" title="MATRIX_1:NM.2">Matrix</a> of 3,<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Maroon" title="c5">N</font> holds <br/><font color="Olive">b<sub>1</sub></font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_6.html#D2" onmouseover="rs('matrix_6/D2')" onmouseout="rh()">MATRIX_6:def 2</a>, <a class="ref" href="matrix_6.html#D3" onmouseover="rs('matrix_6/D3')" onmouseout="rh()">MATRIX_6:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
