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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13">Group_of_Perm</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:10"/><i><font color="Green">E14</font></i>: 
( ex <font color="Olive">b<sub>1</sub></font> being   <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> 2 st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="matrix_2.html#NR1" target="_self">is_transposition</a>  ) &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> 2 st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="matrix_2.html#NR1" target="_self">is_transposition</a>  ) )
 ;<br/>

<a NAME="E2:10"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default">2,1</span><a href="finseq_1.html#K10">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="matrix_7.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E3:10"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default">2,1</span><a href="finseq_1.html#K10">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="matrix_7.html#T8" target="_self">Th8</a>, <a class="txt" href="matrix_7.html#E1:10"><i><font color="Green">E14</font></i></a></i>;<br/>
<a NAME="E4:10"/><i><font color="Green">E17</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13">Group_of_Perm</a> 2</span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_2.html#K11">Permutations</a> 2
 
<b>by </b><i><a class="ref" href="matrix_2.html#D13">MATRIX_2:def 13</a></i>;<br/>
<a NAME="E5:10"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default">1,2</span><a href="finseq_1.html#K10">*&gt;</a></span> or <font color="Maroon">c<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default">2,1</span><a href="finseq_1.html#K10">*&gt;</a></span> )
 
<b>by </b><i><a class="txt" href="matrix_7.html#E4:10"><i><font color="Green">E17</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="matrix_7.html#T3" target="_self">Th3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> 2<b> such that </b><br/><a NAME="E7:10"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="matrix_2.html#NR1" target="_self">is_transposition</a>  )
 <b>by </b><i><a class="txt" href="matrix_7.html#E1:10"><i><font color="Green">E14</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> 2<b> such that </b><br/><a NAME="E9:10"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="matrix_2.html#NR1" target="_self">is_transposition</a>  )
 <b>by </b><i><a class="txt" href="matrix_7.html#E1:10"><i><font color="Green">E14</font></i></a></i>;<br/>
<a NAME="E10:10"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E7:10"><i><font color="Green">E19</font></i></a>, <a class="txt" href="matrix_7.html#E9:10"><i><font color="Green">E20</font></i></a>, <a class="ref" href="matrix_7.html#T9" target="_self">Th9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="matrix_7.html#T8" target="_self">Th8</a>, <a class="txt" href="matrix_7.html#E9:10"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E12:10"/><i><font color="Green">E22</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="matrix_7.html#E3:10"><i><font color="Green">E16</font></i></a>, <a class="txt" href="matrix_7.html#E9:10"><i><font color="Green">E20</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="matrix_7.html#T8" target="_self">Th8</a>, <a class="txt" href="matrix_7.html#E7:10"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E14:10"/><i><font color="Green">E23</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="matrix_7.html#E2:10"><i><font color="Green">E15</font></i></a>, <a class="txt" href="matrix_7.html#E7:10"><i><font color="Green">E19</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E15:10"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
;<br/>
<a NAME="E16:10"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E17:10"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="funct_1.html#T23">FUNCT_1:23</a>, <a class="txt" href="matrix_7.html#E14:10"><i><font color="Green">E23</font></i></a>, <a class="txt" href="matrix_7.html#E12:10"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>hence </b><a NAME="E18:10"/>
<font color="Maroon">c<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default">1,2</span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="matrix_7.html#E5:10"><i><font color="Green">E18</font></i></a>, <a class="txt" href="matrix_7.html#E10:10"><i><font color="Green">E21</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>


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