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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:10"/><i><font color="Green" title="E12">A1</font></i>: 
(  ex <font color="Olive" title="b1">p</font> being   <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> 2 st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Olive" title="b1">p</font> <a href="matrix_2.html#NR1" title="MATRIX_2:NR.1">is_transposition</a>  ) &amp;  ex <font color="Olive" title="b1">q</font> being   <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> 2 st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Olive" title="b1">q</font> <a href="matrix_2.html#NR1" title="MATRIX_2:NR.1">is_transposition</a>  ) )
 ;<br/>

<a NAME="E2:10"/><b>then </b><i><font color="Green" title="E13">A2</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default">2,1</span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="matrix_7.html#T8" target="_self" title="MATRIX_7:th.8">Th8</a></i>;<br/>
<a NAME="E3:10"/><i><font color="Green" title="E14">A3</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default">2,1</span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E1:10"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="matrix_7.html#T8" target="_self" title="MATRIX_7:th.8">Th8</a></i>;<br/>
<a NAME="E4:10"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> 2
 
<b>by </b><i><a class="ref" href="matrix_2.html#D13" title="MATRIX_2:def.13">MATRIX_2:def 13</a></i>;<br/>
<a NAME="E5:10"/><b>then </b><i><font color="Green" title="E15">A4</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default">1,2</span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> or <font color="Maroon" title="c1">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default">2,1</span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="matrix_7.html#T3" target="_self" title="MATRIX_7:th.3">Th3</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">p</font> being    <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> 2<b> such that </b><br/><a NAME="E7:10"/><i><font color="Green" title="E16">A5</font></i>: 
( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c3">p</font> <a href="matrix_2.html#NR1" title="MATRIX_2:NR.1">is_transposition</a>  )
 <b>by </b><i><a class="txt" href="matrix_7.html#E1:10"><i><font color="Green" title="E12">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">q</font> being    <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> 2<b> such that </b><br/><a NAME="E9:10"/><i><font color="Green" title="E17">A6</font></i>: 
( <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c4">q</font> <a href="matrix_2.html#NR1" title="MATRIX_2:NR.1">is_transposition</a>  )
 <b>by </b><i><a class="txt" href="matrix_7.html#E1:10"><i><font color="Green" title="E12">A1</font></i></a></i>;<br/>
<a NAME="E10:10"/><i><font color="Green" title="E18">A7</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E7:10"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="matrix_7.html#E9:10"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="matrix_7.html#T9" target="_self" title="MATRIX_7:th.9">Th9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">q2</font> = <font color="Maroon" title="c4">q</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="txt" href="matrix_7.html#E9:10"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="matrix_7.html#T8" target="_self" title="MATRIX_7:th.8">Th8</a></i>;<br/>
<a NAME="E12:10"/><i><font color="Green" title="E19">A8</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c5">q2</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; <font color="Maroon" title="c5">q2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="matrix_7.html#E3:10"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="matrix_7.html#E9:10"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">p2</font> = <font color="Maroon" title="c3">p</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="txt" href="matrix_7.html#E7:10"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="matrix_7.html#T8" target="_self" title="MATRIX_7:th.8">Th8</a></i>;<br/>
<a NAME="E14:10"/><i><font color="Green" title="E20">A9</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c6">p2</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; <font color="Maroon" title="c6">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="matrix_7.html#E2:10"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="matrix_7.html#E7:10"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E15:10"/><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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p2</font></span>)</span>
 
;<br/>
<a NAME="E16:10"/><b>then </b>
1 <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="c6">p2</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E17:10"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="matrix_7.html#E12:10"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="matrix_7.html#E14:10"><i><font color="Green" title="E20">A9</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<b>hence </b><a NAME="E18:10"/>
<font color="Maroon" title="c1">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default">1,2</span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="matrix_7.html#E5:10"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="matrix_7.html#E10:10"><i><font color="Green" title="E18">A7</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>


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