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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>,   <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>,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:41_1_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">q</font>, <font color="Olive">p</font> being   <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>  ex <font color="Olive">z</font> being   <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> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">q</font>,<font color="Olive">p</font>,<font color="Olive">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be    <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>, <font color="Maroon">p</font> be    <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>;<br/>

<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font>, <font color="Maroon">q</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">z</font> = <font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q</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>take </b>
<font color="Maroon">z</font>
;<br/>

<b>thus </b><a NAME="E3:41_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">q</font>,<font color="Maroon">p</font>,<font color="Maroon">z</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">o</font> being   <a href="binop_1.html#NM2">BinOp</a> of  <a href="matrix_2.html#K12" target="_self">Permutations</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:41_1_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">q</font>, <font color="Olive">p</font> being   <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> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">q</font>,<font color="Olive">p</font>,<font color="Maroon">o</font> <a href="binop_1.html#K2">.</a> <font color="Olive">q</font>,<font color="Olive">p</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>();<br/></i>
<b>take </b>
 <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="matrix_2.html#K12" target="_self">Permutations</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">o</font> #)
;<br/>

<b>thus </b><a NAME="E4:41_1_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="matrix_2.html#K12" target="_self">Permutations</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">o</font> #) <a href="hidden.html#R1">=</a>  <a href="matrix_2.html#K12" target="_self">Permutations</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">q</font>, <font color="Olive">p</font> being   <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> holds  the <a href="group_1.html#U1">mult</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="matrix_2.html#K12" target="_self">Permutations</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">o</font> #) <a href="binop_1.html#K1">.</a> <font color="Olive">q</font>,<font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="partfun1.html#K1">*</a> <font color="Olive">q</font> ) )
 <b>by </b><i/>;<br/>


</div>
