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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">K</font> being  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/> for <font color="Olive" title="b2">R</font> being  <a href="matrixj1.html#NM4" title="MATRIXJ1:NM.4">FinSequence_of_Square-Matrix</a> of <font color="Olive" title="b1">K</font> holds   <a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">R</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c1">n</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="b2">R</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c1">n</font></span><br/><b>let </b><font color="Maroon" title="c2">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">R</font> being  <a href="matrixj1.html#NM4" title="MATRIXJ1:NM.4">FinSequence_of_Square-Matrix</a> of <font color="Maroon" title="c2">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">R</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c1">n</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">R</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c1">n</font></span><br/><b>let </b><font color="Maroon" title="c3">R</font> be   <a href="matrixj1.html#NM4" title="MATRIXJ1:NM.4">FinSequence_of_Square-Matrix</a> of <font color="Maroon" title="c2">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c1">n</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="c3">R</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c1">n</font></span><br/>





<a NAME="E1:83"/><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> <font color="Maroon" title="c3">R</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> <font color="Maroon" title="c3">R</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1"/>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">R</font> or <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">R</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1"><b>suppose </b></a><a NAME="E1:83_1_1"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">R</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="c3">R</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c1">n</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="c3">R</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c1">n</font></span><br/><div class="add"><a NAME="E2:83_1_1"/><b>then </b>
( <font color="Maroon" title="c3">R</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">R</font> &amp; <span class="p1">(<span class="default"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c3">R</font> )
 <b>by </b><i><a class="txt" href="matrixj1.html#E1:83"><i><font color="Green" title="E55">A1</font></i></a>, <a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></i>;<br/><b>hence </b><a NAME="E3:83_1_1"/>
 <a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c1">n</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="c3">R</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_2"><b>suppose </b></a><a NAME="E1:83_1_2"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">R</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="c3">R</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c1">n</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="c3">R</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c1">n</font></span><br/><div class="add"><a NAME="E2:83_1_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/><a NAME="E3:83_1_2"/><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"><a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/><a NAME="E4:83_1_2"/>
<font color="Maroon" title="c3">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="matrixj1.html#K19" title="MATRIXJ1:func.19">/^</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="rfinseq.html#T21" title="RFINSEQ:th.21">RFINSEQ:21</a></i>;<br/><a NAME="E5:83_1_2"/><b>then </b>
 <a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <font color="Maroon" title="c3">R</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="c3">R</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c1">n</font></span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="matrixj1.html#K19" title="MATRIXJ1:func.19">/^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="matrixj1.html#T50" target="_self" title="MATRIXJ1:th.50">Th50</a></i>;<br/><b>hence </b><a NAME="E6:83_1_2"/>
 <a href="matrixj1.html#K22" title="MATRIXJ1:func.22">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="matrixj1.html#K18" title="MATRIXJ1:func.18">|</a> <font color="Maroon" title="c1">n</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="c3">R</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="matrixj1.html#E3:83_1_2"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="finseq_5.html#T26" title="FINSEQ_5:th.26">FINSEQ_5:26</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>

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