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

<span class="kw">defpred </span><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#K15" title="MONOID_0:func.15">GFuncs</a> <font color="Maroon" title="c1">X</font></span>)</span>] <span class="kw">means </span>$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="E1:155_1_1"/><span class="lab"><font color="Green" title="E67">A1</font></span>: 
 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#K15" title="MONOID_0:func.15">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="155_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><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="E2:155_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="monoid_0.html#K15" title="MONOID_0:func.15">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#D40" target="_self" title="MONOID_0:def.40">Def40</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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#K15" title="MONOID_0:func.15">GFuncs</a> <font color="Maroon" title="c1">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">x</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>]</span><br/>

<span class="kw">thus </span><a NAME="E4:155_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:155_1_1"/><span class="lab"><font color="Green" title="E68">A2</font></span>: 
<font color="Maroon">H<sub>1</sub></font>( <a href="monoid_0.html#K15" title="MONOID_0:func.15">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#D40" target="_self" title="MONOID_0:def.40">Def40</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">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#K15" title="MONOID_0:func.15">GFuncs</a> <font color="Maroon" title="c1">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></span>;<br/>
<a NAME="E4:155_1_1"/><span class="lab"><font color="Green" title="E69">A3</font></span>: 
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#K15" title="MONOID_0:func.15">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#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="155_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">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#K15" title="MONOID_0:func.15">GFuncs</a> <font color="Maroon" title="c1">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">y</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c4">y</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:155_1_1_2"/>
( <font color="Maroon" title="c3">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="c4">y</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c1">X</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c4">y</font>]</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c6">g</font> = <font color="Maroon" title="c4">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="E3:155_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#T79" target="_self" title="MONOID_0:th.79">Th79</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:155_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c4">y</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c3">G</font> being  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="monoid_0.html#M5" title="MONOID_0:mode.5">SubStr</a> of  <a href="monoid_0.html#K15" title="MONOID_0:func.15">GFuncs</a> <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E6:155_1_1"/><span class="lab"><font color="Green" title="E70">A4</font></span>: 
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#K15" title="MONOID_0:func.15">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="c3">G</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">f</font>] )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="monoid_0.html#S2" target="_self" title="MONOID_0:sch.2">MONOID_0:sch 2</a>(<a class="txt" href="#E4:155_1_1"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="txt" href="#E1:155_1_1"><span class="lab"><font color="Green" title="E67">A1</font></span></a>);<br/></span>
<a NAME="E7:155_1_1"/>
<font color="Maroon" title="c2">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="c3">G</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:155_1_1"><span class="lab"><font color="Green" title="E70">A4</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">G</font> = <font color="Maroon" title="c3">G</font> as  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a>   <a href="monoid_0.html#M5" title="MONOID_0:mode.5">SubStr</a> of  <a href="monoid_0.html#K15" title="MONOID_0:func.15">GFuncs</a> <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#T80" target="_self" title="MONOID_0:th.80">Th80</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="155_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c5">i</font> = <font color="Maroon" title="c2">f</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:155_1_1"><span class="lab"><font color="Green" title="E70">A4</font></span></a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">l</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">G</font> st <br/>( <font color="Maroon" title="c6">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Olive" title="b2">l</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> )</span><br/><a NAME="E2:155_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#K15" title="MONOID_0:func.15">GFuncs</a> <font color="Maroon" title="c1">X</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#T25" target="_self" title="MONOID_0:th.25">Th25</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">a9</font> = <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c9">b9</font> = <font color="Maroon" title="c7">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#K15" title="MONOID_0:func.15">GFuncs</a> <font color="Maroon" title="c1">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">f</font> = <font color="Maroon" title="c8">a9</font>, <font color="Maroon" title="c11">g</font> = <font color="Maroon" title="c9">b9</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:155_1_1"><span class="lab"><font color="Green" title="E70">A4</font></span></a></span>;<br/><a NAME="E5:155_1_1_3"/><span class="lab"><font color="Green" title="E71">A5</font></span>: 
( <font color="Maroon" title="c5">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="c5">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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T88" title="FUNCT_2:th.88">FUNCT_2:88</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">f9</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#K15" title="MONOID_0:func.15">GFuncs</a> <font color="Maroon" title="c1">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:155_1_1"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></span>;<br/><a NAME="E7:155_1_1_3"/><span class="lab"><font color="Green" title="E72">A6</font></span>: 
( <font color="Maroon" title="c11">g</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> &amp; <font color="Maroon" title="c5">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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">f9</font> = <font color="Maroon" title="c12">f9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:155_1_1"><span class="lab"><font color="Green" title="E70">A4</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">r</font> = <font color="Maroon" title="c13">f9</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c15">l</font> = <font color="Maroon" title="c7">b</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c13">f9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">G</font> ;<br/><span class="kw">take </span><font color="Maroon" title="c16">r</font> = <font color="Maroon" title="c14">r</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">l</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">G</font> st <br/>( <font color="Maroon" title="c6">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c16">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Olive" title="b1">l</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c17">l</font> = <font color="Maroon" title="c15">l</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c16">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c17">l</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> )</span><br/><a NAME="E10:155_1_1_3"/><span class="lab"><font color="Green" title="E73">A7</font></span>: 
( <font color="Maroon" title="c5">i</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c7">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="c7">b</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c5">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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#T79" target="_self" title="MONOID_0:th.79">Th79</a></span>;<br/><a NAME="E11:155_1_1_3"/>
( <font color="Maroon" title="c6">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c13">f9</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">f9</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c6">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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#T79" target="_self" title="MONOID_0:th.79">Th79</a></span>;<br/><span class="kw">hence </span><a NAME="E12:155_1_1_3"/>
( <font color="Maroon" title="c6">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c16">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c17">l</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:155_1_1_3"><span class="lab"><font color="Green" title="E71">A5</font></span></a>, <a class="txt" href="#E7:155_1_1_3"><span class="lab"><font color="Green" title="E72">A6</font></span></a>, <a class="txt" href="#E10:155_1_1_3"><span class="lab"><font color="Green" title="E73">A7</font></span></a>, <a class="txt" href="monoid_0.html#E28"><span class="lab"><font color="Green" title="E15">Lm2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><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="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_1.html#V1" title="GROUP_1:attr.1">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#K15" title="MONOID_0:func.15">GFuncs</a> <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#T12" target="_self" title="MONOID_0:th.12">Th12</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">G</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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#K15" title="MONOID_0:func.15">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> )</span><br/>

<span class="kw">thus </span><a NAME="E11:155_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#K15" title="MONOID_0:func.15">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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:155_1_1"><span class="lab"><font color="Green" title="E70">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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