<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">ITP</font> be    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">ITG</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> <font color="Maroon">n</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:32"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">ITG</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ITP</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 ;<br/>

<a NAME="E2:32"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">ITP</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="matrix_2.html#D11">MATRIX_2:def 11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">qf</font> = <font color="Maroon">ITP</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>, <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> ;<br/>
<a NAME="E4:32"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:32"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">qf</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:32"/><i><font color="Green">E43</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> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="matrix_2.html#D13">MATRIX_2:def 13</a></i>;<br/>
<a NAME="E7:32"/><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">qf</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a>, </i>;<br/>
<a NAME="E8:32"/><i><font color="Green">E45</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">ITP</font> <a href="matrix_7.html#K3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">ITP</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">ITP</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">ITP</font> <a href="matrix_7.html#K3" target="_self">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T61">FUNCT_1:61</a></i>;<br/>
<b>reconsider </b><font color="Maroon">pf</font> = <font color="Maroon">ITP</font> <a href="matrix_7.html#K3" target="_self">"</a>  as   <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> <font color="Maroon">n</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E10:32"/><i><font color="Green">E46</font></i>: 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">ITG</font>,<font color="Maroon">pf</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">ITP</font> <a href="matrix_7.html#K3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">ITP</font>
 
<b>by </b><i>, <a class="ref" href="matrix_2.html#D13">MATRIX_2:def 13</a></i>;<br/>
<a NAME="E11:32"/><i><font color="Green">E47</font></i>: 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">pf</font>,<font color="Maroon">ITG</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ITP</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">ITP</font> <a href="matrix_7.html#K3" target="_self">"</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="matrix_2.html#D13">MATRIX_2:def 13</a></i>;<br/>
<a NAME="E12:32"/><i><font color="Green">E52</font></i>: 
 <a href="newton.html#K4">idseq</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K13">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_2.html#T28">MATRIX_2:28</a></i>;<br/>
<a NAME="E13:32"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">ITG</font> <a href="group_1.html#K1">*</a> <font color="Maroon">pf</font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">ITG</font>,<font color="Maroon">pf</font>
 
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>;<br/>
<a NAME="E14:32"/>
<font color="Maroon">pf</font> <a href="group_1.html#K1">*</a> <font color="Maroon">ITG</font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">pf</font>,<font color="Maroon">ITG</font>
 
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E15:32"/>
<font color="Maroon">ITP</font> <a href="matrix_7.html#K3" target="_self">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">ITG</font> <a href="group_1.html#K3">"</a> 
 <b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a>, <a class="txt" href="matrix_7.html#E1"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="matrix_7.html#T1" target="_self">Th1</a>, , , </i>;<br/>


</div>
