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<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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="rlvect_3.html#V1" title="RLVECT_3:attr.1">linearly-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Olive" title="b2">B</font> holds <br/> ex <font color="Olive" title="b3">M</font> being   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> st <br/>( <font color="Olive" title="b3">M</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  &amp; <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Olive" title="b3">M</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Olive" title="b1">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Olive" title="b2">B</font></span>)</span> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">TRn</font> =  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be   <a href="rlvect_3.html#V1" title="RLVECT_3:attr.1">linearly-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">B</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>,<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> st <br/>( <font color="Olive" title="b1">M</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  &amp; <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Olive" title="b1">M</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:28"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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>,<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> st <br/>( <font color="Olive" title="b1">M</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  &amp; <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Olive" title="b1">M</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">BB</font> =  <a href="matrix13.html#K11" title="MATRIX13:func.11">MX2FinS</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> (<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>,<font color="Maroon" title="c1">n</font>)</span>)</span> as    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c1">n</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <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="matrlin2.html#T45" title="MATRLIN2:th.45">MATRLIN2:45</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">V</font> = <font color="Maroon" title="c1">n</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>;<br/>
<a NAME="E3:28"/>
<font color="Maroon" title="c3">A</font> is   <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="matrtop2.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="matrtop2.html#T7" target="_self" title="MATRTOP2:th.7">Th7</a></span>;<br/>
<a NAME="E4:28"/><span class="kw">then </span>
<font color="Maroon" title="c3">A</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_9.html#T21" title="VECTSP_9:th.21">VECTSP_9:21</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">fA</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E6:28"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">fA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">and </span><br/><a NAME="E7:28"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
<font color="Maroon" title="c7">fA</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T58" title="FINSEQ_4:th.58">FINSEQ_4:58</a></span>;<br/>
<a NAME="E8:28"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">fA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="txt" href="#E7:28"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="pre_poly.html#T19" title="PRE_POLY:th.19">PRE_POLY:19</a></span>;<br/>
<a NAME="E9:28"/>
<font color="Maroon" title="c4">B</font> is   <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="matrtop2.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="matrtop2.html#T7" target="_self" title="MATRTOP2:th.7">Th7</a></span>;<br/>
<a NAME="E10:28"/><span class="kw">then </span>
<font color="Maroon" title="c4">B</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_9.html#T21" title="VECTSP_9:th.21">VECTSP_9:21</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">fB</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E12:28"/><span class="lab"><font color="Green" title="E25">A5</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">fB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font>
 <span class="kw">and </span><br/><a NAME="E13:28"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
<font color="Maroon" title="c8">fB</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T58" title="FINSEQ_4:th.58">FINSEQ_4:58</a></span>;<br/>
<a NAME="E14:28"/><span class="lab"><font color="Green" title="E27">A7</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">fB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:28"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="txt" href="#E13:28"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="ref" href="pre_poly.html#T19" title="PRE_POLY:th.19">PRE_POLY:19</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">fA</font> = <font color="Maroon" title="c7">fA</font>, <font color="Maroon" title="c10">fB</font> = <font color="Maroon" title="c8">fB</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="txt" href="#E12:28"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">MA</font> being   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a><span class="kw"> such that </span><br/><a NAME="E17:28"/><span class="lab"><font color="Green" title="E28">A8</font></span>: 
<font color="Maroon" title="c11">MA</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> 
 <span class="kw">and </span><br/><a NAME="E18:28"/><span class="lab"><font color="Green" title="E29">A9</font></span>: 
<font color="Maroon" title="c11">MA</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">fA</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">fA</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="txt" href="#E7:28"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="matrtop2.html#T19" target="_self" title="MATRTOP2:th.19">Th19</a></span>;<br/>
<a NAME="E19:28"/><span class="lab"><font color="Green" title="E30">A10</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">BB</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">fA</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#D2" title="VECTSP_4:def.2">VECTSP_4:def 2</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">Ma</font> =  <a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c11">MA</font>;<br/>
<a NAME="E20:28"/><span class="lab"><font color="Green" title="E31">A11</font></span>: 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c11">MA</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> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:28"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="ref" href="laplace.html#T34" title="LAPLACE:th.34">LAPLACE:34</a></span>;<br/>
<a NAME="E21:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A12</font></span>: 
 <a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c11">MA</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrtop1.html#T40" title="MATRTOP1:th.40">MATRTOP1:40</a></span>;<br/>
<a NAME="E22:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A13</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c11">MA</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c11">MA</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T33" title="FUNCT_1:th.33">FUNCT_1:33</a></span>;<br/>
<a NAME="E23:28"/><span class="lab"><font color="Green" title="E34">A14</font></span>: 
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c11">MA</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="matrtop2.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="funct_2.html#T52" title="FUNCT_2:th.52">FUNCT_2:52</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c11">MA</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">BB</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">fA</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c11">MA</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">BB</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">fA</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:28"><span class="lab"><font color="Green" title="E32">A12</font></span></a>, <a class="ref" href="funct_1.html#T84" title="FUNCT_1:th.84">FUNCT_1:84</a></span>
<br/>.= 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="txt" href="#E7:28"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="txt" href="#E17:28"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="txt" href="#E18:28"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="ref" href="matrtop2.html#T21" target="_self" title="MATRTOP2:th.21">Th21</a></span>
;<br/>
<a NAME="E25:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A15</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c11">MA</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">BB</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">fB</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E8:28"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="txt" href="#E14:28"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="txt" href="#E23:28"><span class="lab"><font color="Green" title="E34">A14</font></span></a>, <a class="txt" href="#E22:28"><span class="lab"><font color="Green" title="E33">A13</font></span></a>, <a class="txt" href="#E19:28"><span class="lab"><font color="Green" title="E30">A10</font></span></a>, <a class="ref" href="funct_1.html#T77" title="FUNCT_1:th.77">FUNCT_1:77</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">MB</font> being   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a><span class="kw"> such that </span><br/><a NAME="E27:28"/><span class="lab"><font color="Green" title="E36">A16</font></span>: 
<font color="Maroon" title="c13">MB</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> 
 <span class="kw">and </span><br/><a NAME="E28:28"/><span class="lab"><font color="Green" title="E37">A17</font></span>: 
<font color="Maroon" title="c13">MB</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">fB</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">fB</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:28"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="txt" href="#E13:28"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="ref" href="matrtop2.html#T19" target="_self" title="MATRTOP2:th.19">Th19</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">Mb</font> =  <a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c13">MB</font>;<br/>
<a NAME="E29:28"/><span class="lab"><font color="Green" title="E38">A18</font></span>: 
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">mb</font> = <font color="Maroon" title="c13">MB</font> as   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">MA</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a></span>)</span>,<font color="Maroon" title="c1">n</font>,<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="matrix13.html#T1" title="MATRIX13:th.1">MATRIX13:1</a></span>;<br/>
<a NAME="E31:28"/><span class="lab"><font color="Green" title="E39">A19</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c13">MB</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="txt" href="#E29:28"><span class="lab"><font color="Green" title="E38">A18</font></span></a>, <a class="ref" href="matrix13.html#T1" title="MATRIX13:th.1">MATRIX13:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">MM</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c11">MA</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a></span>)</span> <a href="matrix_3.html#K5" title="MATRIX_3:func.5">*</a> <font color="Maroon" title="c15">mb</font> as   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:28"><span class="lab"><font color="Green" title="E38">A18</font></span></a>, <a class="ref" href="matrix13.html#T1" title="MATRIX13:th.1">MATRIX13:1</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c16">MM</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">MM</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  &amp; <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c16">MM</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span> )</span><br/>

<a NAME="E33:28"/>
<font color="Maroon" title="c11">MA</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:28"><span class="lab"><font color="Green" title="E28">A8</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">hence </span><a NAME="E34:28"/>
<font color="Maroon" title="c16">MM</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="txt" href="#E27:28"><span class="lab"><font color="Green" title="E36">A16</font></span></a>, <a class="ref" href="matrix_6.html#T45" title="MATRIX_6:th.45">MATRIX_6:45</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c16">MM</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span></span><br/>

<span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c13">MB</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c11">MA</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> = 
<span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c15">mb</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c11">MA</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:28"><span class="lab"><font color="Green" title="E38">A18</font></span></a>, <a class="ref" href="matrix13.html#T1" title="MATRIX13:th.1">MATRIX13:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c15">mb</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">MA</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:28"><span class="lab"><font color="Green" title="E31">A11</font></span></a>, <a class="ref" href="matrtop1.html#T43" title="MATRTOP1:th.43">MATRTOP1:43</a></span>
<br/>.= 
 <a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">MA</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a></span>)</span> <a href="matrix_3.html#K5" title="MATRIX_3:func.5">*</a> <font color="Maroon" title="c15">mb</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:28"><span class="lab"><font color="Green" title="E38">A18</font></span></a>, <a class="ref" href="matrtop1.html#T32" title="MATRTOP1:th.32">MATRTOP1:32</a></span>
<br/>.= 
 <a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c16">MM</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:28"><span class="lab"><font color="Green" title="E38">A18</font></span></a>, <a class="txt" href="#E31:28"><span class="lab"><font color="Green" title="E39">A19</font></span></a>, <a class="ref" href="matrix13.html#T1" title="MATRIX13:th.1">MATRIX13:1</a></span>
;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c16">MM</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c13">MB</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">BB</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">fB</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:28"><span class="lab"><font color="Green" title="E35">A15</font></span></a>, <a class="ref" href="relat_1.html#T126" title="RELAT_1:th.126">RELAT_1:126</a></span>
<br/>.= 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c4">B</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:28"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="txt" href="#E13:28"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="txt" href="#E27:28"><span class="lab"><font color="Green" title="E36">A16</font></span></a>, <a class="txt" href="#E28:28"><span class="lab"><font color="Green" title="E37">A17</font></span></a>, <a class="ref" href="matrtop2.html#T21" target="_self" title="MATRTOP2:th.21">Th21</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
