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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> is    <a href="topgrp_1.html#M3" target="_self">Homeomorphism</a> of <font color="Maroon">T</font>;<br/>
<b>consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:53_1_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">q</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> iff ( <font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">q</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">f</font> be    <a href="topgrp_1.html#M3" target="_self">Homeomorphism</a> of <font color="Maroon">T</font>;<br/><a NAME="E1:53_1_1_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T12">FUNCT_2:12</a></i>;<br/><b>hence </b><a NAME="E2:53_1_1_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="topgrp_1.html#E2:53_1_1"><i><font color="Green">E30</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>] <b>means </b> ex <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="topgrp_1.html#M3" target="_self">Homeomorphism</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="partfun1.html#K1">*</a> <font color="Olive">f</font> );<br/>
<a NAME="E5:53_1_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>  ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/>

<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font>, <font color="Maroon">y1</font> = <font color="Maroon">y</font> as    <a href="topgrp_1.html#M3" target="_self">Homeomorphism</a> of <font color="Maroon">T</font> <b>by </b><i><a class="txt" href="topgrp_1.html#E2:53_1_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E2:53_1_1_2"/>
<font color="Maroon">y1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">x1</font> is    <a href="topgrp_1.html#M3" target="_self">Homeomorphism</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="topgrp_1.html#T2" target="_self">Th2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">y1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">x1</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">z</font>
;<br/>

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

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

<b>thus </b><a NAME="E4:53_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">x1</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 <font color="Maroon">A</font><b> such that </b><br/><a NAME="E7:53_1_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Maroon">o</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">b</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>();<br/></i>
<b>take </b><font color="Maroon">G</font> =  <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">A</font>,<font color="Maroon">o</font> #);<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E8:53_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> iff <font color="Maroon">x</font> is    <a href="topgrp_1.html#M3" target="_self">Homeomorphism</a> of <font color="Maroon">T</font> )
 <b>by </b><i><a class="txt" href="topgrp_1.html#E2:53_1_1"><i><font color="Green">E30</font></i></a>, </i>;<br/>

<b>let </b><font color="Maroon">f</font> be    <a href="topgrp_1.html#M3" target="_self">Homeomorphism</a> of <font color="Maroon">T</font>, <font color="Maroon">g</font> be    <a href="topgrp_1.html#M3" target="_self">Homeomorphism</a> of <font color="Maroon">T</font>;<br/>

<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font>, <font color="Maroon">g1</font> = <font color="Maroon">g</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<a NAME="E10:53_1_1"/>
 ex <font color="Olive">m</font>, <font color="Olive">n</font> being   <a href="topgrp_1.html#M3" target="_self">Homeomorphism</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> &amp; <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp; <font color="Maroon">o</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">f1</font>,<font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> <a href="partfun1.html#K1">*</a> <font color="Olive">m</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E11:53_1_1"/>
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
 ;<br/>


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