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<div class="add">

<b>let </b><font color="Maroon" title="c1">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">R1</font>, <font color="Olive" title="b2">R2</font> being  <a href="matrixj1.html#NM4" title="MATRIXJ1:NM.4">FinSequence_of_Square-Matrix</a> of <font color="Maroon" title="c1">K</font> holds   <a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">R1</font> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <font color="Olive" title="b2">R2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Olive" title="b1">R1</font></span>)</span> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Olive" title="b2">R2</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">R1</font>, <font color="Maroon" title="c3">R2</font> be   <a href="matrixj1.html#NM4" title="MATRIXJ1:NM.4">FinSequence_of_Square-Matrix</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R1</font> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <font color="Maroon" title="c3">R2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c2">R1</font></span>)</span> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c3">R2</font></span>)</span></span><br/>





<b>set </b><font color="Maroon" title="c4">R12</font> = <font color="Maroon" title="c2">R1</font> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <font color="Maroon" title="c3">R2</font>;<br/>
<a NAME="E1:82"/><i><font color="Green" title="E55">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">R1</font> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <font color="Maroon" title="c3">R2</font></span>)</span></span>)</span> <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="c2">R1</font> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <font color="Maroon" title="c3">R2</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R1</font> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <font color="Maroon" title="c3">R2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">R1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">R2</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">R1</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"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c2">R1</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">R2</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"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c3">R2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a>, <a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E2:82"/><b>then </b><i><font color="Green" title="E56">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c2">R1</font></span>)</span> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c3">R2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">R1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">R2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E3:82"/><b>then </b><i><font color="Green" title="E57">A3</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c2">R1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">R1</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c3">R2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">R2</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c2">R1</font></span>)</span> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c3">R2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">R1</font> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <font color="Maroon" title="c3">R2</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="matrixj1.html#E1:82"><i><font color="Green" title="E55">A1</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> &amp; <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">R1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">R2</font></span>)</span> implies <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">R1</font> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <font color="Maroon" title="c3">R2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c2">R1</font></span>)</span> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c3">R2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> )</span><br/><b>assume </b><a NAME="E1:82_1"/><i><font color="Green" title="E58">A4</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> &amp; <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">R1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">R2</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">R1</font> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <font color="Maroon" title="c3">R2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c2">R1</font></span>)</span> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c3">R2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span><br/><a NAME="E2:82_1"/><i><font color="Green" title="E59">A5</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c2">R1</font></span>)</span> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c3">R2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrixj1.html#E2:82"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="matrixj1.html#E1:82_1"><i><font color="Green" title="E58">A4</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:82_1_1_1"/>
( <font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c2">R1</font></span>)</span> or  ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c3">R2</font></span>)</span> &amp; <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c2">R1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font> ) )
 </span><b>by </b><i><a class="txt" href="matrixj1.html#E2:82_1"><i><font color="Green" title="E59">A5</font></i></a>, <a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_1_1"><b>suppose </b></a><a NAME="E1:82_1_1_1_1"/><i><font color="Green" title="E60">A6</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c2">R1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c2">R1</font></span>)</span> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c3">R2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">R1</font> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <font color="Maroon" title="c3">R2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span><br/><div class="add"><a NAME="E2:82_1_1_1_1"/><i><font color="Green" title="E61">A7</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R1</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">R1</font> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <font color="Maroon" title="c3">R2</font></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c5">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrixj1.html#E3:82"><i><font color="Green" title="E57">A3</font></i></a>, <a class="txt" href="matrixj1.html#E1:82_1_1_1_1"><i><font color="Green" title="E60">A6</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c2">R1</font></span>)</span> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c3">R2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> = 
<span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c2">R1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
<b>by </b><i><a class="txt" href="matrixj1.html#E1:82_1_1_1_1"><i><font color="Green" title="E60">A6</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R1</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c5">k</font></span>)</span>
<b>by </b><i><a class="txt" href="matrixj1.html#E1:82_1_1_1_1"><i><font color="Green" title="E60">A6</font></i></a>, <a class="ref" href="matrixj1.html#D7" target="_self" title="MATRIXJ1:def.7">Def7</a></i>
<br/>.= 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">R1</font> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <font color="Maroon" title="c3">R2</font></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c5">k</font></span>)</span>
<b>by </b><i><a class="txt" href="matrixj1.html#E3:82"><i><font color="Green" title="E57">A3</font></i></a>, <a class="txt" href="matrixj1.html#E1:82_1_1_1_1"><i><font color="Green" title="E60">A6</font></i></a>, <a class="txt" href="matrixj1.html#E2:82_1_1_1_1"><i><font color="Green" title="E61">A7</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">R1</font> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <font color="Maroon" title="c3">R2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
<b>by </b><i><a class="txt" href="matrixj1.html#E3:82"><i><font color="Green" title="E57">A3</font></i></a>, <a class="txt" href="matrixj1.html#E2:82_1"><i><font color="Green" title="E59">A5</font></i></a>, <a class="ref" href="matrixj1.html#D7" target="_self" title="MATRIXJ1:def.7">Def7</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_1_2"><b>suppose </b></a><a NAME="E1:82_1_1_1_2"/>
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c3">R2</font></span>)</span> &amp; <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c2">R1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c2">R1</font></span>)</span> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c3">R2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">R1</font> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <font color="Maroon" title="c3">R2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:82_1_1_1_2"/><i><font color="Green" title="E60">A8</font></i>: 
( <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c3">R2</font></span>)</span> &amp; <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">R1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n</font> )
 <b>by </b><i><a class="txt" href="matrixj1.html#E1:82"><i><font color="Green" title="E55">A1</font></i></a></i>;<br/><a NAME="E4:82_1_1_1_2"/><i><font color="Green" title="E61">A9</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R2</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">R1</font> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <font color="Maroon" title="c3">R2</font></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c5">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrixj1.html#E3:82"><i><font color="Green" title="E57">A3</font></i></a>, <a class="txt" href="matrixj1.html#E3:82_1_1_1_2"><i><font color="Green" title="E60">A8</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c2">R1</font></span>)</span> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c3">R2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> = 
<span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c3">R2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>
<b>by </b><i><a class="txt" href="matrixj1.html#E1:82"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="matrixj1.html#E3:82_1_1_1_2"><i><font color="Green" title="E60">A8</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R2</font> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
<b>by </b><i><a class="txt" href="matrixj1.html#E3:82_1_1_1_2"><i><font color="Green" title="E60">A8</font></i></a>, <a class="ref" href="matrixj1.html#D7" target="_self" title="MATRIXJ1:def.7">Def7</a></i>
<br/>.= 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">R1</font> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <font color="Maroon" title="c3">R2</font></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c5">k</font></span>)</span>
<b>by </b><i><a class="txt" href="matrixj1.html#E4:82_1_1_1_2"><i><font color="Green" title="E61">A9</font></i></a>, <a class="txt" href="matrixj1.html#E3:82"><i><font color="Green" title="E57">A3</font></i></a>, <a class="txt" href="matrixj1.html#E3:82_1_1_1_2"><i><font color="Green" title="E60">A8</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">R1</font> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <font color="Maroon" title="c3">R2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
<b>by </b><i><a class="txt" href="matrixj1.html#E3:82"><i><font color="Green" title="E57">A3</font></i></a>, <a class="txt" href="matrixj1.html#E2:82_1"><i><font color="Green" title="E59">A5</font></i></a>, <a class="ref" href="matrixj1.html#D7" target="_self" title="MATRIXJ1:def.7">Def7</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:82_1"/>
<span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">R1</font> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <font color="Maroon" title="c3">R2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c2">R1</font></span>)</span> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c3">R2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:82"/>
 <a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R1</font> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <font color="Maroon" title="c3">R2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c2">R1</font></span>)</span> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c3">R2</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrixj1.html#E1:82"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="matrixj1.html#E2:82"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
