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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"> ( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 implies  ex <font color="Olive" title="b1">ODD</font>, <font color="Olive" title="b2">EVEN</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">EVEN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">p</font> where <font color="Olive" title="b3">p</font> is    <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> : <font color="Olive" title="b3">p</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> </span>}</span>  &amp; <font color="Olive" title="b1">ODD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">q</font> where <font color="Olive" title="b3">q</font> is    <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> : not <font color="Olive" title="b3">q</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> </span>}</span>  &amp; <font color="Olive" title="b2">EVEN</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b1">ODD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b2">EVEN</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b1">ODD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">EVEN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">ODD</font> ) )</span><br/>

<b>assume </b><a NAME="E1:66"/><i><font color="Green" title="E57">A1</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">ODD</font>, <font color="Olive" title="b2">EVEN</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">EVEN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">p</font> where <font color="Olive" title="b3">p</font> is    <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> : <font color="Olive" title="b3">p</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> </span>}</span>  &amp; <font color="Olive" title="b1">ODD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">q</font> where <font color="Olive" title="b3">q</font> is    <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> : not <font color="Olive" title="b3">q</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> </span>}</span>  &amp; <font color="Olive" title="b2">EVEN</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b1">ODD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b2">EVEN</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b1">ODD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">EVEN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">ODD</font> )</span><br/>

<a NAME="E2:66"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="#E1:66"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:66"/><b>then </b>
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="txt" href="#E1:66"><i><font color="Green" title="E57">A1</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c2">O</font>, <font color="Maroon" title="c3">E</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:66"/><i><font color="Green" title="E58">A2</font></i>: 
( <font color="Maroon" title="c3">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is    <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> : <font color="Olive" title="b1">p</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> </span>}</span>  &amp; <font color="Maroon" title="c2">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is    <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> : not <font color="Olive" title="b1">q</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> </span>}</span>  )
 <b>and </b><br/><a NAME="E6:66"/><i><font color="Green" title="E59">A3</font></i>: 
( <font color="Maroon" title="c3">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">E</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> )
 <b>and </b><br/><a NAME="E7:66"/><i><font color="Green" title="E60">A4</font></i>: 
 ex <font color="Olive" title="b1">P</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">E</font>,<font color="Maroon" title="c2">O</font> ex <font color="Olive" title="b2">perm</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> st <br/>( <font color="Olive" title="b2">perm</font> is <a href="matrix_2.html#V4" title="MATRIX_2:attr.4">being_transposition</a> &amp; <font color="Olive" title="b2">perm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">E</font> &amp; <font color="Olive" title="b1">P</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> &amp; ( for <font color="Olive" title="b3">p</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>  st <font color="Olive" title="b3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font> holds <br/><font color="Olive" title="b1">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">perm</font> ) )
 <b>by </b><i><a class="txt" href="matrix11.html#E59"><i><font color="Green" title="E56">Lm8</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">P</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">E</font>,<font color="Maroon" title="c2">O</font>, <font color="Maroon" title="c5">perm</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><b> such that </b><br/><a NAME="E9:66"/><i><font color="Green" title="E61">A5</font></i>: 
( <font color="Maroon" title="c5">perm</font> is <a href="matrix_2.html#V4" title="MATRIX_2:attr.4">being_transposition</a> &amp; <font color="Maroon" title="c5">perm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">E</font> )
 <b>and </b><br/><a NAME="E10:66"/><i><font color="Green" title="E62">A6</font></i>: 
( <font color="Maroon" title="c4">P</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> &amp; ( for <font color="Olive" title="b1">p</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>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font> holds <br/><font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">perm</font> ) )
 <b>by </b><i><a class="txt" href="#E7:66"><i><font color="Green" title="E60">A4</font></i></a></i>;<br/>
<a NAME="E11:66"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">O</font> &amp; <font color="Maroon" title="c4">P</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 
<b>by </b><i><a class="txt" href="#E10:66"><i><font color="Green" title="E62">A6</font></i></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E12:66"/><b>then </b>
<font color="Maroon" title="c3">E</font>,<font color="Maroon" title="c2">O</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="#E9:66"><i><font color="Green" title="E61">A5</font></i></a>, <a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></i>;<br/>
<a NAME="E13:66"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">O</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>;<br/>
<b>hence </b><a NAME="E14:66"/>
 ex <font color="Olive" title="b1">ODD</font>, <font color="Olive" title="b2">EVEN</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">EVEN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">p</font> where <font color="Olive" title="b3">p</font> is    <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> : <font color="Olive" title="b3">p</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> </span>}</span>  &amp; <font color="Olive" title="b1">ODD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">q</font> where <font color="Olive" title="b3">q</font> is    <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> : not <font color="Olive" title="b3">q</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> </span>}</span>  &amp; <font color="Olive" title="b2">EVEN</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b1">ODD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b2">EVEN</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b1">ODD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">EVEN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">ODD</font> )
 <b>by </b><i><a class="txt" href="#E5:66"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="#E6:66"><i><font color="Green" title="E59">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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