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

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="group_1.html#V1">strict</a>  <a href="group_1.html#L1">HGrStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:48_1_2"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> implies <font color="Olive">b<sub>1</sub></font> is    <a href="topgrp_1.html#M1" target="_self">Homeomorphism</a> of <font color="Maroon">c<sub>1</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> is    <a href="topgrp_1.html#M1" target="_self">Homeomorphism</a> of <font color="Maroon">c<sub>1</sub></font> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> ) &amp; ( for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="topgrp_1.html#M1" target="_self">Homeomorphism</a> of <font color="Maroon">c<sub>1</sub></font> holds the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font> ) )
 <b>and </b><br/><a NAME="E2:48_1_2"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> implies <font color="Olive">b<sub>1</sub></font> is    <a href="topgrp_1.html#M1" target="_self">Homeomorphism</a> of <font color="Maroon">c<sub>1</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> is    <a href="topgrp_1.html#M1" target="_self">Homeomorphism</a> of <font color="Maroon">c<sub>1</sub></font> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> ) &amp; ( for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="topgrp_1.html#M1" target="_self">Homeomorphism</a> of <font color="Maroon">c<sub>1</sub></font> holds the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>1</sub></font> is    <a href="topgrp_1.html#M1" target="_self">Homeomorphism</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E3:48_1_2"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] )
 
<b>by </b><i><a class="txt" href="topgrp_1.html#E1:48_1_2"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E4:48_1_2"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] )
 
<b>by </b><i><a class="txt" href="topgrp_1.html#E2:48_1_2"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E5:48_1_2"/><i><font color="Green">E32</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>from </b><i><a class="ref" href="xboole_0.html#S2">XBOOLE_0:sch 2</a>(<a class="txt" href="topgrp_1.html#E3:48_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="topgrp_1.html#E4:48_1_2"><i><font color="Green">E31</font></i></a>);<br/></i>
<a NAME="E6:48_1_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> holds <br/>the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:48_1_2_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="topgrp_1.html#M1" target="_self">Homeomorphism</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="topgrp_1.html#E1:48_1_2"><i><font color="Green">E28</font></i></a></i>;<br/>
<b>thus </b>the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="topgrp_1.html#E1:48_1_2"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="topgrp_1.html#E2:48_1_2"><i><font color="Green">E29</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:48_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="topgrp_1.html#E5:48_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="binop_1.html#T1">BINOP_1:1</a></i>;<br/>


</div>
