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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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> <font color="Maroon" title="c1">n</font> holds <br/> ( ( ( <font color="Olive" title="b1">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; <font color="Olive" title="b2">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  ) or ( not <font color="Olive" title="b1">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; not <font color="Olive" title="b2">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  ) ) iff <font color="Olive" title="b1">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</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> <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( ( <font color="Maroon" title="c2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; <font color="Maroon" title="c3">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  ) or ( not <font color="Maroon" title="c2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; not <font color="Maroon" title="c3">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">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#V5" title="MATRIX_2:attr.5">even</a>  )</span><br/>





<span class="kw">reconsider </span><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#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_9.html#T39" title="MATRIX_9:th.39">MATRIX_9:39</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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"><span class="kw">suppose </span></a><a NAME="E1:34_1_1_1"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( ( <font color="Maroon" title="c2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; <font color="Maroon" title="c3">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  ) or ( not <font color="Maroon" title="c2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; not <font color="Maroon" title="c3">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">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#V5" title="MATRIX_2:attr.5">even</a>  )</span><br/><div class="add"><a NAME="E2:34_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">pq</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="matrix11.html#E29"><span class="lab"><font color="Green" title="E28">Lm3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:34_1_1_1"/>
( ( ( <font color="Maroon" title="c2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; <font color="Maroon" title="c3">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  ) or ( not <font color="Maroon" title="c2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; not <font color="Maroon" title="c3">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">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#V5" title="MATRIX_2:attr.5">even</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1_1_1"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="matrix11.html#E29"><span class="lab"><font color="Green" title="E28">Lm3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_2"><span class="kw">suppose </span></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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( ( ( <font color="Maroon" title="c2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; <font color="Maroon" title="c3">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  ) or ( not <font color="Maroon" title="c2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; not <font color="Maroon" title="c3">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">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#V5" title="MATRIX_2:attr.5">even</a>  ) &amp; (  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#V5" title="MATRIX_2:attr.5">even</a>  or ( <font color="Maroon" title="c2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; <font color="Maroon" title="c3">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  ) or ( not <font color="Maroon" title="c2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; not <font color="Maroon" title="c3">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  ) ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c6">K</font> being   <a href="vectsp_1.html#V12" title="VECTSP_1:attr.12">Fanoian</a>  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">p9</font> = <font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c8">q9</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#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="c5">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> ;<br/><span class="kw">thus </span><a NAME="E5:34_1_1_2"/>
( ( ( <font color="Maroon" title="c2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; <font color="Maroon" title="c3">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  ) or ( not <font color="Maroon" title="c2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; not <font color="Maroon" title="c3">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">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#V5" title="MATRIX_2:attr.5">even</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  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#V5" title="MATRIX_2:attr.5">even</a>  or ( <font color="Maroon" title="c2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; <font color="Maroon" title="c3">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  ) or ( not <font color="Maroon" title="c2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; not <font color="Maroon" title="c3">q</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="34_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:34_1_1_2_1"/>
( ( <font color="Maroon" title="c2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; <font color="Maroon" title="c3">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  ) or ( not <font color="Maroon" title="c2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; not <font color="Maroon" title="c3">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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#V5" title="MATRIX_2:attr.5">even</a> </span><br/>

<a NAME="E2:34_1_1_2_1"/><span class="kw">then </span>
( (  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c7">p9</font>,<font color="Maroon" title="c6">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="c6">K</font> &amp;  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c8">q9</font>,<font color="Maroon" title="c6">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="c6">K</font> ) or (  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c7">p9</font>,<font color="Maroon" title="c6">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="c6">K</font></span>)</span> &amp;  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c8">q9</font>,<font color="Maroon" title="c6">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="c6">K</font></span>)</span> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix11.html#T23" target="_self" title="MATRIX11:th.23">Th23</a></span>;<br/>
<a NAME="E3:34_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c7">p9</font>,<font color="Maroon" title="c6">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c8">q9</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#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c6">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c6">K</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T42" title="VECTSP_1:th.42">VECTSP_1:42</a></span>;<br/>
<a NAME="E4:34_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c6">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">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#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c6">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></span>;<br/>
<a NAME="E5:34_1_1_2_1"/><span class="kw">then </span>
 <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#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c6">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_1_1_2_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="matrix11.html#T24" target="_self" title="MATRIX11:th.24">Th24</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:34_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#V5" title="MATRIX_2:attr.5">even</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix11.html#T23" target="_self" title="MATRIX11:th.23">Th23</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><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#V5" title="MATRIX_2:attr.5">even</a>  or ( <font color="Maroon" title="c2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; <font color="Maroon" title="c3">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  ) or ( not <font color="Maroon" title="c2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; not <font color="Maroon" title="c3">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><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#V5" title="MATRIX_2:attr.5">even</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; <font color="Maroon" title="c3">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  ) or ( not <font color="Maroon" title="c2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; not <font color="Maroon" title="c3">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  ) )</span><br/>

<a NAME="E2:34_1_1_2_2"/><span class="kw">then </span>
 <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#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c6">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix11.html#T23" target="_self" title="MATRIX11:th.23">Th23</a></span>;<br/>
<a NAME="E3:34_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A3</font></span>: 
<span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c7">p9</font>,<font color="Maroon" title="c6">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c8">q9</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#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c6">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix11.html#T24" target="_self" title="MATRIX11:th.24">Th24</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:34_1_1_2_2"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon" title="c2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; <font color="Maroon" title="c3">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  ) &amp; not <!--HUMANRECFAILED-->( not <font color="Maroon" title="c2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; not <font color="Maroon" title="c3">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_2_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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#V5" title="MATRIX_2:attr.5">even</a>  &amp; not <font color="Maroon" title="c3">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  ) or ( not <font color="Maroon" title="c2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; <font color="Maroon" title="c3">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_1_1_2_2"><span class="lab"><font color="Green" title="E31">A4</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_2_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_1_1_2_2_1_1_1"/><span class="lab"><font color="Green" title="E32">A5</font></span>: 
( <font color="Maroon" title="c2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; not <font color="Maroon" title="c3">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:34_1_1_2_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A6</font></span>: 
 <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c8">q9</font>,<font color="Maroon" title="c6">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="c6">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix11.html#T23" target="_self" title="MATRIX11:th.23">Th23</a></span>;<br/><a NAME="E3:34_1_1_2_2_1_1_1"/>
 <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c7">p9</font>,<font color="Maroon" title="c6">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="c6">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="matrix11.html#T23" target="_self" title="MATRIX11:th.23">Th23</a></span>;<br/><a NAME="E4:34_1_1_2_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c7">p9</font>,<font color="Maroon" title="c6">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c8">q9</font>,<font color="Maroon" title="c6">K</font></span>)</span> <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="c6">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></span>;<br/><span class="kw">hence </span><a NAME="E5:34_1_1_2_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_1_1_2_2"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="matrix11.html#T22" target="_self" title="MATRIX11:th.22">Th22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_2_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_1_1_2_2_1_1_2"/><span class="lab"><font color="Green" title="E32">A7</font></span>: 
( not <font color="Maroon" title="c2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; <font color="Maroon" title="c3">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:34_1_1_2_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A8</font></span>: 
 <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c8">q9</font>,<font color="Maroon" title="c6">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="c6">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix11.html#T23" target="_self" title="MATRIX11:th.23">Th23</a></span>;<br/><a NAME="E3:34_1_1_2_2_1_1_2"/>
 <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c7">p9</font>,<font color="Maroon" title="c6">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="c6">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="ref" href="matrix11.html#T23" target="_self" title="MATRIX11:th.23">Th23</a></span>;<br/><a NAME="E4:34_1_1_2_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c7">p9</font>,<font color="Maroon" title="c6">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c8">q9</font>,<font color="Maroon" title="c6">K</font></span>)</span> <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="c6">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_1_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></span>;<br/><span class="kw">hence </span><a NAME="E5:34_1_1_2_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_1_1_2_2"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="matrix11.html#T22" target="_self" title="MATRIX11:th.22">Th22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:34_1_1_2_2"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:34"/>
( ( ( <font color="Maroon" title="c2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; <font color="Maroon" title="c3">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  ) or ( not <font color="Maroon" title="c2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; not <font color="Maroon" title="c3">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">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#V5" title="MATRIX_2:attr.5">even</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
