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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1 is    <a href="topgrp_1.html#M3" title="TOPGRP_1:mode.3">Homeomorphism</a> of <font color="Maroon" title="c1">T</font>;<br/>
<b>consider </b><font color="Maroon" title="c2">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:52_1_1"/><i><font color="Green" title="E30">A1</font></i>: 
for <font color="Olive" title="b1">q</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> iff ( <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">q</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<div><i><font color="Green" title="E31">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">f</font> be    <a href="topgrp_1.html#M3" title="TOPGRP_1:mode.3">Homeomorphism</a> of <font color="Maroon" title="c1">T</font>;<br/><a NAME="E1:52_1_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></i>;<br/><b>hence </b><a NAME="E2:52_1_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="topgrp_1.html#E2:52_1_1"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c3">A</font> = <font color="Maroon" title="c2">A</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font>] <b>means </b> ex <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="topgrp_1.html#M3" title="TOPGRP_1:mode.3">Homeomorphism</a> of <font color="Maroon" title="c1">T</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">f</font> );<br/>
<a NAME="E5:52_1_1"/><i><font color="Green" title="E32">A3</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font>  ex <font color="Olive" title="b3">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c6">x1</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c7">y1</font> = <font color="Maroon" title="c5">y</font> as    <a href="topgrp_1.html#M3" title="TOPGRP_1:mode.3">Homeomorphism</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="topgrp_1.html#E2:52_1_1"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/>
<a NAME="E2:52_1_1_2"/>
<font color="Maroon" title="c7">y1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">x1</font> is    <a href="topgrp_1.html#M3" title="TOPGRP_1:mode.3">Homeomorphism</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="topgrp_1.html#T31" target="_self" title="TOPGRP_1:th.31">Th31</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">z</font> = <font color="Maroon" title="c7">y1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">x1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> <b>by </b><i><a class="txt" href="topgrp_1.html#E3:52_1_1"><i><font color="Green" title="E31">A2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">z</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c6">x1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c7">y1</font>
;<br/>

<b>thus </b><a NAME="E4:52_1_1_2"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x1</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y1</font> &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">x1</font> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">o</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c3">A</font><b> such that </b><br/><a NAME="E7:52_1_1"/><i><font color="Green" title="E33">A4</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Maroon" title="c4">o</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="topgrp_1.html#E5:52_1_1"><i><font color="Green" title="E32">A3</font></i></a>);<br/></i>
<b>take </b><font color="Maroon" title="c5">G</font> =  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">o</font> #);<br/>

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E8:52_1_1"/>
( <font color="Maroon" title="c6">x</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="Maroon" title="c6">x</font> is    <a href="topgrp_1.html#M3" title="TOPGRP_1:mode.3">Homeomorphism</a> of <font color="Maroon" title="c1">T</font> )
 <b>by </b><i><a class="txt" href="topgrp_1.html#E2:52_1_1"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="topgrp_1.html#E3:52_1_1"><i><font color="Green" title="E31">A2</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c7">f</font>, <font color="Maroon" title="c8">g</font> be    <a href="topgrp_1.html#M3" title="TOPGRP_1:mode.3">Homeomorphism</a> of <font color="Maroon" title="c1">T</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c9">f1</font> = <font color="Maroon" title="c7">f</font>, <font color="Maroon" title="c10">g1</font> = <font color="Maroon" title="c8">g</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> <b>by </b><i><a class="txt" href="topgrp_1.html#E3:52_1_1"><i><font color="Green" title="E31">A2</font></i></a></i>;<br/>
<a NAME="E10:52_1_1"/>
 ex <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being   <a href="topgrp_1.html#M3" title="TOPGRP_1:mode.3">Homeomorphism</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Maroon" title="c9">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Maroon" title="c10">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Maroon" title="c4">o</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">f1</font>,<font color="Maroon" title="c10">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">n</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">m</font> )
 
<b>by </b><i><a class="txt" href="topgrp_1.html#E7:52_1_1"><i><font color="Green" title="E33">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:52_1_1"/>
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c5">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">f</font>
 ;<br/>


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