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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K16" title="MONOID_0:func.16">GFuncs</a> <font color="Maroon" title="c1">X</font></span>)</span>] <b>means </b>$1 is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c1">X</font>;<br/>
<a NAME="E1:148_1_1"/><i><font color="Green" title="E66">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K16" title="MONOID_0:func.16">GFuncs</a> <font color="Maroon" title="c1">X</font></span>)</span>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">y</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="148_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K16" title="MONOID_0:func.16">GFuncs</a> <font color="Maroon" title="c1">X</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:148_1_1_1"/>
( <font color="Maroon" title="c2">x</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c3">y</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c1">X</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c3">y</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E3:148_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">g</font>
 
<b>by </b><i><a class="ref" href="monoid_0.html#T79" target="_self" title="MONOID_0:th.79">Th79</a></i>;<br/>
<b>hence </b><a NAME="E4:148_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <font color="Maroon" title="c3">y</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:148_1_1"/><i><font color="Green" title="E67">A2</font></i>: 
 ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K16" title="MONOID_0:func.16">GFuncs</a> <font color="Maroon" title="c1">X</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="148_1_1_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c2">f</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c1">X</font>;<br/>
<a NAME="E2:148_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="monoid_0.html#K16" title="MONOID_0:func.16">GFuncs</a> <font color="Maroon" title="c1">X</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="monoid_0.html#D40" target="_self" title="MONOID_0:def.40">Def40</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">x</font> = <font color="Maroon" title="c2">f</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K16" title="MONOID_0:func.16">GFuncs</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">x</font>
;<br/>

<b>thus </b><a NAME="E4:148_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c2">G</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="monoid_0.html#M5" title="MONOID_0:mode.5">SubStr</a> of  <a href="monoid_0.html#K16" title="MONOID_0:func.16">GFuncs</a> <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:148_1_1"/><i><font color="Green" title="E68">A3</font></i>: 
for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K16" title="MONOID_0:func.16">GFuncs</a> <font color="Maroon" title="c1">X</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">f</font>] )
 <b>from </b><i><a class="ref" href="monoid_0.html#S2" target="_self" title="MONOID_0:sch.2">MONOID_0:sch 2</a>(<a class="txt" href="monoid_0.html#E1:148_1_1"><i><font color="Green" title="E66">A1</font></i></a>, <a class="txt" href="monoid_0.html#E2:148_1_1"><i><font color="Green" title="E67">A2</font></i></a>);<br/></i>
<a NAME="E5:148_1_1"/><i><font color="Green" title="E69">A4</font></i>: 
<font color="Maroon">H<sub>1</sub></font>( <a href="monoid_0.html#K16" title="MONOID_0:func.16">GFuncs</a> <font color="Maroon" title="c1">X</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="monoid_0.html#D40" target="_self" title="MONOID_0:def.40">Def40</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">f</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">X</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K16" title="MONOID_0:func.16">GFuncs</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></i>;<br/>
<a NAME="E7:148_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">G</font>)
 
<b>by </b><i><a class="txt" href="monoid_0.html#E4:148_1_1"><i><font color="Green" title="E68">A3</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">G</font> = <font color="Maroon" title="c2">G</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a> <a href="group_1.html#V2" title="GROUP_1:attr.2">unital</a>  <a href="monoid_0.html#M5" title="MONOID_0:mode.5">SubStr</a> of  <a href="monoid_0.html#K16" title="MONOID_0:func.16">GFuncs</a> <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="monoid_0.html#T80" target="_self" title="MONOID_0:th.80">Th80</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="148_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">G</font>;<br/><a NAME="E1:148_1_1_3"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">G</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">H<sub>1</sub></font>( <a href="monoid_0.html#K16" title="MONOID_0:func.16">GFuncs</a> <font color="Maroon" title="c1">X</font>)
 <b>by </b><i><a class="ref" href="monoid_0.html#T25" target="_self" title="MONOID_0:th.25">Th25</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">a'</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c8">b'</font> = <font color="Maroon" title="c6">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K16" title="MONOID_0:func.16">GFuncs</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">i</font> = <font color="Maroon" title="c3">f</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">G</font> <b>by </b><i><a class="txt" href="monoid_0.html#E4:148_1_1"><i><font color="Green" title="E68">A3</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">f</font> = <font color="Maroon" title="c7">a'</font>, <font color="Maroon" title="c11">g</font> = <font color="Maroon" title="c8">b'</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="monoid_0.html#E4:148_1_1"><i><font color="Green" title="E68">A3</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">f'</font> = <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K16" title="MONOID_0:func.16">GFuncs</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="txt" href="monoid_0.html#E5:148_1_1"><i><font color="Green" title="E69">A4</font></i></a>, <a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">f'</font> = <font color="Maroon" title="c12">f'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">G</font> <b>by </b><i><a class="txt" href="monoid_0.html#E4:148_1_1"><i><font color="Green" title="E68">A3</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">r</font> = <font color="Maroon" title="c13">f'</font> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c15">l</font> = <font color="Maroon" title="c6">b</font> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <font color="Maroon" title="c13">f'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">G</font> ;<br/><b>take </b><font color="Maroon" title="c16">r</font> = <font color="Maroon" title="c14">r</font>;<br/><b>take </b><font color="Maroon" title="c17">l</font> = <font color="Maroon" title="c15">l</font>;<br/><a NAME="E8:148_1_1_3"/>
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> &amp; <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c10">f</font> &amp; <font color="Maroon" title="c11">g</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> &amp; <font color="Maroon" title="c9">i</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> &amp; <font color="Maroon" title="c5">a</font> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <font color="Maroon" title="c13">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> &amp; <font color="Maroon" title="c13">f'</font> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c10">f</font> &amp; <font color="Maroon" title="c9">i</font> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c11">g</font> &amp; <font color="Maroon" title="c6">b</font> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">X</font></span>)</span> )
 <b>by </b><i><a class="ref" href="monoid_0.html#T79" target="_self" title="MONOID_0:th.79">Th79</a>, <a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a>, <a class="ref" href="funct_2.html#T88" title="FUNCT_2:th.88">FUNCT_2:88</a></i>;<br/><b>hence </b><a NAME="E9:148_1_1_3"/>
( <font color="Maroon" title="c5">a</font> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <font color="Maroon" title="c16">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c17">l</font> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="txt" href="monoid_0.html#E28"><i><font color="Green" title="E14">Lm2</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">G</font> = <font color="Maroon" title="c4">G</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a> <a href="group_1.html#V2" title="GROUP_1:attr.2">unital</a> <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a>  <a href="monoid_0.html#M5" title="MONOID_0:mode.5">SubStr</a> of  <a href="monoid_0.html#K16" title="MONOID_0:func.16">GFuncs</a> <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="monoid_0.html#T12" target="_self" title="MONOID_0:th.12">Th12</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">G</font>
;<br/>

<b>thus </b><a NAME="E11:148_1_1"/>
for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K16" title="MONOID_0:func.16">GFuncs</a> <font color="Maroon" title="c1">X</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G</font> iff <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c1">X</font> )
 <b>by </b><i><a class="txt" href="monoid_0.html#E4:148_1_1"><i><font color="Green" title="E68">A3</font></i></a></i>;<br/>


</div>
