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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>;<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>;<br/><b>let </b><font color="Maroon" title="c3">M</font> be   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>;<br/><b>let </b><font color="Maroon" title="c4">perm</font> be    <a href="matrix_2.html#M3" title="MATRIX_2:mode.3">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font>;<br/><b>let </b><font color="Maroon" title="c5">Perm</font> be   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>;<br/>









<b>assume </b><a NAME="E1:63"/><i><font color="Green" title="E55">A1</font></i>: 
<font color="Maroon" title="c5">Perm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">perm</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_1"/>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 or <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1"><b>suppose </b></a><a NAME="E1:63_1_1"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2
 ;<br/><div class="add"><a NAME="E2:63_1_1"/><b>then </b>
( <font color="Maroon" title="c4">perm</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> &amp; <font color="Maroon" title="c4">perm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="matrix11.html#E29"><i><font color="Green" title="E28">Lm3</font></i></a></i>;<br/><a NAME="E3:63_1_1"/><b>then </b>
( <font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">perm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span>,<font color="Maroon" title="c4">perm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font> )
 <b>by </b><i><a class="ref" href="matrix11.html#T39" target="_self" title="MATRIX11:th.39">Th39</a>, <a class="ref" href="matrix_2.html#D16" title="MATRIX_2:def.16">MATRIX_2:def 16</a></i>;<br/><b>hence </b><a NAME="E4:63_1_1"/>
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c5">Perm</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span>,<font color="Maroon" title="c4">perm</font>
 <b>by </b><i><a class="txt" href="matrix11.html#E1:63"><i><font color="Green" title="E55">A1</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2"><b>suppose </b></a><a NAME="E1:63_1_2"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c6">n2</font> = <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">M'</font> = <font color="Maroon" title="c3">M</font> as   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c6">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2,<font color="Maroon" title="c2">K</font> ;<br/><b>reconsider </b><font color="Maroon" title="c8">Perm2</font> = <font color="Maroon" title="c5">Perm</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> ;<br/><b>reconsider </b><font color="Maroon" title="c9">perm2</font> = <font color="Maroon" title="c4">perm</font> as    <a href="matrix_2.html#M3" title="MATRIX_2:mode.3">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> ;<br/><a NAME="E6:63_1_2"/>
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">M'</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c8">Perm2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c9">perm2</font>,<font color="Maroon" title="c2">K</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c7">M'</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix11.html#E1:63"><i><font color="Green" title="E55">A1</font></i></a>, <a class="ref" href="matrix11.html#T45" target="_self" title="MATRIX11:th.45">Th45</a></i>;<br/><b>hence </b><a NAME="E7:63_1_2"/>
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c5">Perm</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span>,<font color="Maroon" title="c4">perm</font>
 <b>by </b><i><a class="ref" href="matrix11.html#T26" target="_self" title="MATRIX11:th.26">Th26</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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