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<b>let </b><font color="Maroon">p</font> be   <a href="funct_2.html#NM2">Permutation</a> of  <a href="setwop_2.html#K2">Seg</a> 3;<br/>

<b>assume </b><a NAME="E1:44"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">2,3,1</span><a href="finseq_4.html#K3">*&gt;</a></span>
 ;<br/>

<a NAME="E2:44"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default">1,2,3</span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T30">FINSEQ_3:30</a>, <a class="ref" href="finseq_3.html#T1">FINSEQ_3:1</a></i>;<br/>
<a NAME="E3:44"/>
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  holds <br/>(  not <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> or  not <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> or  not <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">j</font> or  not <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">j</font> or  not <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> or  ex <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">i</font> &amp; <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">j</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> &amp; not <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">i</font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">j</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E1:44_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; ( for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> &amp; <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> ) )
 ;<br/>

<a NAME="E2:44_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> &amp; <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:44_1_1"/><i><font color="Green">E38</font></i>: 
( ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 3 ) &amp; ( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3 ) )
 
<b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a>, <a class="txt" href="matrix_9.html#E1:44_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="matrix_9.html#E1:44_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_1_1_1"/>
( ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2 ) or ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3 ) or ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3 ) or ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2 ) )
 </span><b>by </b><i>, <a class="txt" href="matrix_9.html#E1:44_1_1"><i><font color="Green">E38</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_1"><b>suppose </b></a><a NAME="E1:44_1_1_1_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2 )
 ;<br/><div class="add"><b>take </b>
3
;<br/><b>thus </b><a NAME="E2:44_1_1_1_1"/>
( 3 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> &amp; 3 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> &amp; 3 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="txt" href="matrix_9.html#E1:44_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_2"><b>suppose </b></a><a NAME="E1:44_1_1_1_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3 )
 ;<br/><div class="add"><b>take </b>
1
;<br/><b>thus </b><a NAME="E2:44_1_1_1_2"/>
( 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> &amp; 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="txt" href="matrix_9.html#E1:44_1_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="matrix_9.html#E1:44_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_3"><b>suppose </b></a><a NAME="E1:44_1_1_1_3"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3 )
 ;<br/><div class="add"><b>take </b>
2
;<br/><b>thus </b><a NAME="E2:44_1_1_1_3"/>
( 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> &amp; 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> &amp; 2 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="txt" href="matrix_9.html#E1:44_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_4"><b>suppose </b></a><a NAME="E1:44_1_1_1_4"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><b>take </b>
3
;<br/><b>thus </b><a NAME="E2:44_1_1_1_4"/>
( 3 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> &amp; 3 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> &amp; 3 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="txt" href="matrix_9.html#E1:44_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_5"><b>suppose </b></a><a NAME="E1:44_1_1_1_5"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><b>take </b>
2
;<br/><b>thus </b><a NAME="E2:44_1_1_1_5"/>
( 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> &amp; 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> &amp; 2 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="txt" href="matrix_9.html#E1:44_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_6"><b>suppose </b></a><a NAME="E1:44_1_1_1_6"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 3 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2 )
 ;<br/><div class="add"><b>take </b>
1
;<br/><b>thus </b><a NAME="E2:44_1_1_1_6"/>
( 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> &amp; 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="txt" href="matrix_9.html#E1:44_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:44_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> )
 ;<br/>
<a NAME="E5:44_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_1_2"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 3 )
 </span><b>by </b><i><a class="txt" href="matrix_9.html#E1:44_1"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_2_1"><b>suppose </b></a><a NAME="E1:44_1_2_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:44_1_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_2_2"><b>suppose </b></a><a NAME="E1:44_1_2_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:44_1_2_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_2_3"><b>suppose </b></a><a NAME="E1:44_1_2_3"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:44_1_2_3"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:44"/>
not <font color="Maroon">p</font> <a href="matrix_2.html#NR1" target="_self">is_transposition</a> 
 <b>by </b><i><a class="ref" href="matrix_2.html#D14">MATRIX_2:def 14</a></i>;<br/>


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