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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">perm2</font> being   <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span><br/> for <font color="Olive" title="b2">K</font> being <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">Fanoian</a> <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a> holds <br/> ( ( <font color="Olive" title="b1">perm2</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> implies  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Olive" title="b1">perm2</font>,<font color="Olive" title="b2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Olive" title="b2">K</font> ) &amp; (  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Olive" title="b1">perm2</font>,<font color="Olive" title="b2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Olive" title="b2">K</font> implies <font color="Olive" title="b1">perm2</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> ) &amp; ( not <font color="Olive" title="b1">perm2</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> implies  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Olive" title="b1">perm2</font>,<font color="Olive" title="b2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Olive" title="b2">K</font></span>)</span> ) &amp; (  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Olive" title="b1">perm2</font>,<font color="Olive" title="b2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Olive" title="b2">K</font></span>)</span> implies  not <font color="Olive" title="b1">perm2</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> ) )</span><br/><b>let </b><font color="Maroon" title="c2">perm2</font> be    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>; <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#V9" title="VECTSP_1:attr.9">Fanoian</a> <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a> holds <br/> ( ( <font color="Maroon" title="c2">perm2</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> implies  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c2">perm2</font>,<font color="Olive" title="b1">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Olive" title="b1">K</font> ) &amp; (  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c2">perm2</font>,<font color="Olive" title="b1">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Olive" title="b1">K</font> implies <font color="Maroon" title="c2">perm2</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> ) &amp; ( not <font color="Maroon" title="c2">perm2</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> implies  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c2">perm2</font>,<font color="Olive" title="b1">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Olive" title="b1">K</font></span>)</span> ) &amp; (  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c2">perm2</font>,<font color="Olive" title="b1">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Olive" title="b1">K</font></span>)</span> implies  not <font color="Maroon" title="c2">perm2</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> ) )</span><br/>



<b>set </b><font color="Maroon" title="c3">n2</font> = <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2;<br/>
<b>let </b><font color="Maroon" title="c4">K</font> be  <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">Fanoian</a> <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"> ( ( <font color="Maroon" title="c2">perm2</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> implies  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c2">perm2</font>,<font color="Maroon" title="c4">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">K</font> ) &amp; (  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c2">perm2</font>,<font color="Maroon" title="c4">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">K</font> implies <font color="Maroon" title="c2">perm2</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> ) &amp; ( not <font color="Maroon" title="c2">perm2</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> implies  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c2">perm2</font>,<font color="Maroon" title="c4">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">K</font></span>)</span> ) &amp; (  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c2">perm2</font>,<font color="Maroon" title="c4">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">K</font></span>)</span> implies  not <font color="Maroon" title="c2">perm2</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> ) )</span><br/>

<a NAME="E1:31"/><i><font color="Green" title="E27">A1</font></i>: 
 <a href="matrix_2.html#K11" title="MATRIX_2:func.11">len</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<b>by </b><i><a class="ref" href="matrix_2.html#T20" title="MATRIX_2:th.20">MATRIX_2:20</a></i>;<br/>
<b>thus </b><a NAME="E2:31"/><i><font color="Green" title="E28">A2</font></i>: 
( <font color="Maroon" title="c2">perm2</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> implies  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c2">perm2</font>,<font color="Maroon" title="c4">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">K</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( (  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c2">perm2</font>,<font color="Maroon" title="c4">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">K</font> implies <font color="Maroon" title="c2">perm2</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> ) &amp; ( not <font color="Maroon" title="c2">perm2</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> implies  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c2">perm2</font>,<font color="Maroon" title="c4">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">K</font></span>)</span> ) &amp; (  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c2">perm2</font>,<font color="Maroon" title="c4">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">K</font></span>)</span> implies  not <font color="Maroon" title="c2">perm2</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1"/>
<font color="Maroon" title="c2">perm2</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c2">perm2</font>,<font color="Maroon" title="c4">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">K</font></span><br/>

<a NAME="E2:31_1"/><b>then </b>
 ex <font color="Olive" title="b1">L</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> st <br/>( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">L</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c2">perm2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Olive" title="b1">L</font> &amp; ( for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">i</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> <font color="Olive" title="b1">L</font> holds <br/> ex <font color="Olive" title="b3">q2</font> being   <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">q2</font> &amp; <font color="Olive" title="b3">q2</font> <a href="matrix_2.html#NR1" title="MATRIX_2:NR.1">is_transposition</a>  ) ) )
 
<b>by </b><i><a class="txt" href="matrix11.html#E1:31"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="matrix_2.html#D15" title="MATRIX_2:def.15">MATRIX_2:def 15</a></i>;<br/>
<b>hence </b><a NAME="E3:31_1"/>
 <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c2">perm2</font>,<font color="Maroon" title="c4">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">K</font>
 <b>by </b><i><a class="ref" href="matrix11.html#T15" target="_self" title="MATRIX11:th.15">Th15</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>

<b>thus </b><a NAME="E3:31"/>
(  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c2">perm2</font>,<font color="Maroon" title="c4">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">K</font> implies <font color="Maroon" title="c2">perm2</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c2">perm2</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> iff  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c2">perm2</font>,<font color="Maroon" title="c4">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">K</font></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_2"/><i><font color="Green" title="E29">A3</font></i>: 
 <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c2">perm2</font>,<font color="Maroon" title="c4">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">perm2</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a></span><br/>

<b>consider </b><font color="Maroon" title="c5">P</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:31_2"/><i><font color="Green" title="E30">A4</font></i>: 
<font color="Maroon" title="c2">perm2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c5">P</font>
 <b>and </b><br/><a NAME="E4:31_2"/><i><font color="Green" title="E31">A5</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</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> <font color="Maroon" title="c5">P</font> holds <br/> ex <font color="Olive" title="b2">trans</font> being   <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">trans</font> &amp; <font color="Olive" title="b2">trans</font> <a href="matrix_2.html#NR1" title="MATRIX_2:NR.1">is_transposition</a>  )
 <b>by </b><i><a class="ref" href="matrix11.html#T21" target="_self" title="MATRIX11:th.21">Th21</a></i>;<br/>
<b>assume </b><a NAME="E5:31_2"/>
not <font color="Maroon" title="c2">perm2</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:31_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="txt" href="matrix11.html#E1:31"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="matrix11.html#E3:31_2"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="matrix11.html#E4:31_2"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="matrix_2.html#D15" title="MATRIX_2:def.15">MATRIX_2:def 15</a></i>;<br/>
<a NAME="E7:31_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="nat_d.html#T12" title="NAT_D:th.12">NAT_D:12</a></i>;<br/>
<a NAME="E8:31_2"/><b>then </b>
 <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c2">perm2</font>,<font color="Maroon" title="c4">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">K</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix11.html#E3:31_2"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="matrix11.html#E4:31_2"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="matrix11.html#T15" target="_self" title="MATRIX11:th.15">Th15</a></i>;<br/>
<b>hence </b><a NAME="E9:31_2"/>
contradiction
 <b>by </b><i><a class="txt" href="matrix11.html#E1:31_2"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="matrix11.html#T22" target="_self" title="MATRIX11:th.22">Th22</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>

<b>hence </b><a NAME="E4:31"/>
( not <font color="Maroon" title="c2">perm2</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> iff  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c2">perm2</font>,<font color="Maroon" title="c4">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">K</font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix11.html#E2:31"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="matrix11.html#T11" target="_self" title="MATRIX11:th.11">Th11</a>, <a class="ref" href="matrix11.html#T22" target="_self" title="MATRIX11:th.22">Th22</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>
