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

<a NAME="E1:48_1_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="group_1.html#K1">*</a> <font color="Olive">q</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">q</font> <a href="group_1.html#K1">*</a> <font color="Olive">r</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">r</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font>, <font color="Maroon">r'</font> = <font color="Maroon">r</font> as    <a href="matrix_2.html#M3" target="_self">Element</a> of  <a href="matrix_2.html#K12" target="_self">Permutations</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<a NAME="E2:48_1_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">q'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">p'</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp; <font color="Maroon">r'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q'</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p'</font>, <font color="Maroon">q'</font> = <font color="Maroon">q'</font>, <font color="Maroon">r'</font> = <font color="Maroon">r'</font> as   <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/>;<br/>
<a NAME="E2:48_1_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">q'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">p'</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
;<br/>
<a NAME="E3:48_1_1_1_1"/>
<font color="Maroon">r'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q'</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
;<br/>
<b>hence </b><a NAME="E4:48_1_1_1_1"/>
( <font color="Maroon">q'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">p'</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp; <font color="Maroon">r'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q'</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/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:48_1_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">q'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">p'</font> is    <a href="matrix_2.html#M3" target="_self">Element</a> of  <a href="matrix_2.html#K12" target="_self">Permutations</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:48_1_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">r'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q'</font> is    <a href="matrix_2.html#M3" target="_self">Element</a> of  <a href="matrix_2.html#K12" target="_self">Permutations</a> <font color="Maroon">n</font>
 
<b>by </b><i>, </i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="group_1.html#K1">*</a> <font color="Maroon">q</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">r</font> = 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="group_1.html#K1">*</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <span class="p2">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font></span>)</span>,<font color="Maroon">r'</font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">q'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">p'</font></span>)</span>,<font color="Maroon">r'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">r'</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">q'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">p'</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q'</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">p'</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">p'</font>,<span class="p1">(<span class="default"><font color="Maroon">r'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q'</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <span class="p2">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">q</font>,<font color="Maroon">r</font></span>)</span>
<b>by </b><i/>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="group_1.html#K1">*</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="group_1.html#K1">*</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E6:48_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="group_1.html#K1">*</a> <font color="Maroon">q</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="group_1.html#K1">*</a> <font color="Maroon">r</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:48_1_1"/>
 ex <font color="Olive">e</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span> st <br/>for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span> holds <br/> ( <font color="Olive">p</font> <a href="group_1.html#K1">*</a> <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Olive">e</font> <a href="group_1.html#K1">*</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp;  ex <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span> st <br/>( <font color="Olive">p</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> &amp; <font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">e'</font> =  <a href="finseq_2.html#K1">idseq</a> <font color="Maroon">n</font> as    <a href="matrix_2.html#M3" target="_self">Element</a> of  <a href="matrix_2.html#K12" target="_self">Permutations</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">e</font> =  <a href="finseq_2.html#K1">idseq</a> <font color="Maroon">n</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E3:48_1_1_2"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span> holds <br/> ( <font color="Olive">p</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as    <a href="matrix_2.html#M3" target="_self">Element</a> of  <a href="matrix_2.html#K12" target="_self">Permutations</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<i><font color="Green">E63</font></i>: <font color="Maroon">p</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font> = 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">p'</font>,<font color="Maroon">e'</font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
<font color="Maroon">e'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">p'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">p</font>
<b>by </b><i/>
;<br/>
<font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Maroon">p</font> = 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">e'</font>,<font color="Maroon">p'</font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
<font color="Maroon">p'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">e'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">p</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E4:48_1_1_2_1"/>
( <font color="Maroon">p</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">e</font>
;<br/>

<a NAME="E4:48_1_1_2"/>
for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span>  ex <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span> st <br/>( <font color="Olive">p</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">p</font> as    <a href="matrix_2.html#M3" target="_self">Element</a> of  <a href="matrix_2.html#K12" target="_self">Permutations</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">q</font> as   <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/>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">r</font> <a href="funct_2.html#K2">"</a>  as    <a href="matrix_2.html#M3" target="_self">Element</a> of  <a href="matrix_2.html#K12" target="_self">Permutations</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i/>;<br/>
<i><font color="Green">E64</font></i>: <font color="Maroon">p</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font> = 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">q</font>,<font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i/>
;<br/>
<i><font color="Green">E68</font></i>: <font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">p</font> = 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">q</font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i/>
;<br/>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>thus </b><a NAME="E7:48_1_1_2_2"/>
( <font color="Maroon">p</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:48_1_1_2"/>
for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span> holds <br/> ( <font color="Olive">p</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp;  ex <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span> st <br/>( <font color="Olive">p</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:48_1_1"/>
(  <a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font> is <a href="group_1.html#V4">associative</a> &amp;  <a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">n</font> is <a href="group_1.html#V3">Group-like</a> )
 <b>by </b><i>, <a class="ref" href="group_1.html#T5">GROUP_1:5</a></i>;<br/>


</div>
