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

<b>set </b><font color="Maroon">G</font> =  <a href="topgrp_1.html#K5" target="_self">HomeoGroup</a> <font color="Maroon">T</font>;<br/>
<b>set </b><font color="Maroon">o</font> = the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="topgrp_1.html#K5" target="_self">HomeoGroup</a> <font color="Maroon">T</font></span>)</span>;<br/>
<b>thus </b><a NAME="E1:55_1_1"/>
 <a href="topgrp_1.html#K5" target="_self">HomeoGroup</a> <font color="Maroon">T</font> is <a href="group_1.html#V3">Group-like</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">e</font> =  <a href="topgrp_1.html#K4" target="_self">id</a> <font color="Maroon">T</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topgrp_1.html#K5" target="_self">HomeoGroup</a> <font color="Maroon">T</font></span>)</span> <b>by </b><i><a class="txt" href="topgrp_1.html#E9:55_1_1_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">e</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D3">GROUP_1:def 3</a><br/>

<b>let </b><font color="Maroon">h</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topgrp_1.html#K5" target="_self">HomeoGroup</a> <font color="Maroon">T</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">h1</font> = <font color="Maroon">h</font>, <font color="Maroon">e1</font> = <font color="Maroon">e</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#E9:55_1_1_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font> = 
<font color="Maroon">e1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h1</font>
<b>by </b><i><a class="txt" href="topgrp_1.html#E9:55_1_1_1"><i><font color="Green">E30</font></i></a></i>
<br/>.= 
<font color="Maroon">h</font>
<b>by </b><i><a class="ref" href="tmap_1.html#T93">TMAP_1:93</a></i>
;<br/>

<b>thus </b><font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> = 
<font color="Maroon">h1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">e1</font>
<b>by </b><i><a class="txt" href="topgrp_1.html#E9:55_1_1_1"><i><font color="Green">E30</font></i></a></i>
<br/>.= 
<font color="Maroon">h</font>
<b>by </b><i><a class="ref" href="tmap_1.html#T93">TMAP_1:93</a></i>
;<br/>

<b>reconsider </b><font color="Maroon">h1</font> = <font color="Maroon">h</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#E9:55_1_1_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E6:55_1_1_1"/>
<font color="Maroon">h1</font> <a href="tops_2.html#K2">/"</a>  is    <a href="topgrp_1.html#M3" target="_self">Homeomorphism</a> of <font color="Maroon">T</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">h1</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#K5" target="_self">HomeoGroup</a> <font color="Maroon">T</font></span>)</span> <b>by </b><i><a class="txt" href="topgrp_1.html#E9:55_1_1_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">g</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#E9:55_1_1_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E9:55_1_1_1"/><i><font color="Green">E30</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</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">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font> = 
<font color="Maroon">g1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h1</font>
<b>by </b><i><a class="txt" href="topgrp_1.html#E9:55_1_1_1"><i><font color="Green">E30</font></i></a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i>, <a class="ref" href="tops_2.html#T65">TOPS_2:65</a></i>
;<br/>

<b>thus </b><font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> = 
<font color="Maroon">h1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g1</font>
<b>by </b><i><a class="txt" href="topgrp_1.html#E9:55_1_1_1"><i><font color="Green">E30</font></i></a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i>, <a class="ref" href="tops_2.html#T65">TOPS_2:65</a></i>
;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topgrp_1.html#K5" target="_self">HomeoGroup</a> <font color="Maroon">T</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topgrp_1.html#K5" target="_self">HomeoGroup</a> <font color="Maroon">T</font></span>)</span>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topgrp_1.html#K5" target="_self">HomeoGroup</a> <font color="Maroon">T</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a><br/>

<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">x</font>, <font color="Maroon">g</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/>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>, <font color="Maroon">r</font> = <font color="Maroon">z</font> as    <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>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">r</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font> as    <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>thus </b><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font> = 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="topgrp_1.html#K5" target="_self">HomeoGroup</a> <font color="Maroon">T</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">z</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">r</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">p</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="topgrp_1.html#K5" target="_self">HomeoGroup</a> <font color="Maroon">T</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">a</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i/>
;<br/>


</div>
