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

<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">p</font>, <font color="Maroon" title="c3">q</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>reconsider </b><font color="Maroon" title="c4">pq</font> = <font color="Maroon" title="c2">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">q</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> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="matrix_9.html#T39" title="MATRIX_9:th.39">MATRIX_9:39</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_1_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="34_1_1_1"><b>suppose </b></a><a NAME="E1:34_1_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="E2:34_1_1_1"/><b>then </b>
( <font color="Maroon" title="c2">p</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> &amp; <font color="Maroon" title="c3">q</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> &amp; <font color="Maroon" title="c4">pq</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> )
 <b>by </b><i><a class="txt" href="matrix11.html#E29"><i><font color="Green" title="E28">Lm3</font></i></a></i>;<br/><b>hence </b><a NAME="E3:34_1_1_1"/>
( ( ( <font color="Maroon" title="c2">p</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> &amp; <font color="Maroon" title="c3">q</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> ) or ( not <font color="Maroon" title="c2">p</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> &amp; not <font color="Maroon" title="c3">q</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> ) ) iff <font color="Maroon" title="c2">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">q</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_2"><b>suppose </b></a><a NAME="E1:34_1_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="c5">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>consider </b><font color="Maroon" title="c6">K</font> being  <a href="vectsp_1.html#V13" title="VECTSP_1:attr.13">Fanoian</a> <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>reconsider </b><font color="Maroon" title="c7">p'</font> = <font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c8">q'</font> = <font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c9">pq</font> = <font color="Maroon" title="c4">pq</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="c5">n2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2</span>)</span> ;<br/><b>thus </b><a NAME="E5:34_1_1_2"/>
( ( ( <font color="Maroon" title="c2">p</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> &amp; <font color="Maroon" title="c3">q</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> ) or ( not <font color="Maroon" title="c2">p</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> &amp; not <font color="Maroon" title="c3">q</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> ) ) implies <font color="Maroon" title="c2">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">q</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_1_1_2_1"/>
( ( <font color="Maroon" title="c2">p</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> &amp; <font color="Maroon" title="c3">q</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> ) or ( not <font color="Maroon" title="c2">p</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> &amp; not <font color="Maroon" title="c3">q</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> ) )
 ;<br/>

<a NAME="E2:34_1_1_2_1"/><b>then </b>
( (  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c6">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c6">K</font> &amp;  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c8">q'</font>,<font color="Maroon" title="c6">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c6">K</font> ) or (  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c6">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c6">K</font></span>)</span> &amp;  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c8">q'</font>,<font color="Maroon" title="c6">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c6">K</font></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="matrix11.html#T23" target="_self" title="MATRIX11:th.23">Th23</a></i>;<br/>
<a NAME="E3:34_1_1_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c6">K</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c8">q'</font>,<font color="Maroon" title="c6">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c6">K</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c6">K</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c6">K</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c6">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c6">K</font> )
 
<b>by </b><i><a class="ref" href="vectsp_1.html#T42" title="VECTSP_1:th.42">VECTSP_1:42</a>, <a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></i>;<br/>
<a NAME="E4:34_1_1_2_1"/><b>then </b>
 <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c9">pq</font>,<font color="Maroon" title="c6">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c6">K</font>
 
<b>by </b><i><a class="ref" href="matrix11.html#T24" target="_self" title="MATRIX11:th.24">Th24</a></i>;<br/>
<b>hence </b><a NAME="E5:34_1_1_2_1"/>
<font color="Maroon" title="c2">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">q</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a>
 <b>by </b><i><a class="ref" href="matrix11.html#T23" target="_self" title="MATRIX11:th.23">Th23</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E6:34_1_1_2"/>
(  not <font color="Maroon" title="c2">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">q</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> or ( <font color="Maroon" title="c2">p</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> &amp; <font color="Maroon" title="c3">q</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> ) or ( not <font color="Maroon" title="c2">p</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> &amp; not <font color="Maroon" title="c3">q</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_1_1_2_2"/>
<font color="Maroon" title="c2">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">q</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a>
 ;<br/>

<a NAME="E2:34_1_1_2_2"/><b>then </b>
 <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c9">pq</font>,<font color="Maroon" title="c6">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c6">K</font>
 
<b>by </b><i><a class="ref" href="matrix11.html#T23" target="_self" title="MATRIX11:th.23">Th23</a></i>;<br/>
<a NAME="E3:34_1_1_2_2"/><b>then </b><i><font color="Green" title="E30">A1</font></i>: 
<span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c6">K</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c8">q'</font>,<font color="Maroon" title="c6">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c6">K</font>
 
<b>by </b><i><a class="ref" href="matrix11.html#T24" target="_self" title="MATRIX11:th.24">Th24</a></i>;<br/>
<b>assume </b><a NAME="E4:34_1_1_2_2"/><i><font color="Green" title="E31">A2</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon" title="c2">p</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> &amp; <font color="Maroon" title="c3">q</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> ) &amp; not <!--HUMANRECFAILED-->( not <font color="Maroon" title="c2">p</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> &amp; not <font color="Maroon" title="c3">q</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_1_1_2_2_1_1"/>
( ( <font color="Maroon" title="c2">p</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> &amp; not <font color="Maroon" title="c3">q</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> ) or ( not <font color="Maroon" title="c2">p</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> &amp; <font color="Maroon" title="c3">q</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> ) )
 </span><b>by </b><i><a class="txt" href="matrix11.html#E4:34_1_1_2_2"><i><font color="Green" title="E31">A2</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:34_1_1_2_2_1_1_1"/>
( <font color="Maroon" title="c2">p</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> &amp; not <font color="Maroon" title="c3">q</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> )
 ;<br/><div class="add"><a NAME="E2:34_1_1_2_2_1_1_1"/><b>then </b>
(  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c6">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c6">K</font> &amp;  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c8">q'</font>,<font color="Maroon" title="c6">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c6">K</font></span>)</span> )
 <b>by </b><i><a class="ref" href="matrix11.html#T23" target="_self" title="MATRIX11:th.23">Th23</a></i>;<br/><a NAME="E3:34_1_1_2_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c6">K</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c8">q'</font>,<font color="Maroon" title="c6">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c6">K</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></i>;<br/><b>hence </b><a NAME="E4:34_1_1_2_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="matrix11.html#E3:34_1_1_2_2"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="matrix11.html#T22" target="_self" title="MATRIX11:th.22">Th22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:34_1_1_2_2_1_1_2"/>
( not <font color="Maroon" title="c2">p</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> &amp; <font color="Maroon" title="c3">q</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> )
 ;<br/><div class="add"><a NAME="E2:34_1_1_2_2_1_1_2"/><b>then </b>
(  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c6">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c6">K</font></span>)</span> &amp;  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c8">q'</font>,<font color="Maroon" title="c6">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c6">K</font> )
 <b>by </b><i><a class="ref" href="matrix11.html#T23" target="_self" title="MATRIX11:th.23">Th23</a></i>;<br/><a NAME="E3:34_1_1_2_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c7">p'</font>,<font color="Maroon" title="c6">K</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c8">q'</font>,<font color="Maroon" title="c6">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c6">K</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></i>;<br/><b>hence </b><a NAME="E4:34_1_1_2_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="matrix11.html#E3:34_1_1_2_2"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="matrix11.html#T22" target="_self" title="MATRIX11:th.22">Th22</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:34_1_1_2_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:34"/>
( ( ( <font color="Maroon" title="c2">p</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> &amp; <font color="Maroon" title="c3">q</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> ) or ( not <font color="Maroon" title="c2">p</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> &amp; not <font color="Maroon" title="c3">q</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> ) ) iff <font color="Maroon" title="c2">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">q</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> )
 ;<br/>


</div>
