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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="topgrp_1.html#M1" target="_self">Homeomorphism</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topgrp_1.html#K4" target="_self">HomeoGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:52"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="topgrp_1.html#K4" target="_self">HomeoGroup</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E2:52"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tops_2.html#K2">/"</a>  is    <a href="topgrp_1.html#M1" target="_self">Homeomorphism</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:52_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tops_2.html#K2">/"</a>  <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="topgrp_1.html#D3" target="_self">TOPGRP_1:def 3</a><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="tops_2.html#K2">/"</a>  as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topgrp_1.html#K4" target="_self">HomeoGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="topgrp_1.html#D4" target="_self">Def4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>now </b></a><div class="add"><a NAME="E1:52_2"/><i><font color="Green">E32</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> = 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="topgrp_1.html#K4" target="_self">HomeoGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="tops_2.html#K2">/"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="topgrp_1.html#E1:52"><i><font color="Green">E30</font></i></a>, <a class="txt" href="topgrp_1.html#E2:52"><i><font color="Green">E31</font></i></a>, <a class="ref" href="topgrp_1.html#D4" target="_self">Def4</a></i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topgrp_1.html#E1:52_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tops_2.html#T65">TOPS_2:65</a></i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>

<br/>.= 
 <a href="topgrp_1.html#K3" target="_self">id</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="grcat_1.html#D11">GRCAT_1:def 11</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="topgrp_1.html#K4" target="_self">HomeoGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="topgrp_1.html#T33" target="_self">Th33</a></i>
;<br/><b>thus </b><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> = 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="topgrp_1.html#K4" target="_self">HomeoGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>

<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="tops_2.html#K2">/"</a> </span>)</span>
<b>by </b><i><a class="txt" href="topgrp_1.html#E1:52"><i><font color="Green">E30</font></i></a>, <a class="txt" href="topgrp_1.html#E2:52"><i><font color="Green">E31</font></i></a>, <a class="ref" href="topgrp_1.html#D4" target="_self">Def4</a></i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topgrp_1.html#E1:52_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tops_2.html#T65">TOPS_2:65</a></i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>

<br/>.= 
 <a href="topgrp_1.html#K3" target="_self">id</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="grcat_1.html#D11">GRCAT_1:def 11</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="topgrp_1.html#K4" target="_self">HomeoGroup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="topgrp_1.html#T33" target="_self">Th33</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:52"/>
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="tops_2.html#K2">/"</a> 
 <b>by </b><i><a class="ref" href="group_1.html#T12">GROUP_1:12</a></i>;<br/>


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