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<b>let </b><font color="Maroon">A</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_1.html#L1">HGrStr</a> , <font color="Maroon">B</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="E1:53_1_2"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> implies <font color="Olive">x</font> is    <a href="topgrp_1.html#M3" target="_self">Homeomorphism</a> of <font color="Maroon">T</font> ) &amp; ( <font color="Olive">x</font> is    <a href="topgrp_1.html#M3" target="_self">Homeomorphism</a> of <font color="Maroon">T</font> implies <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> ) &amp; ( for <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> holds  the <a href="group_1.html#U1">mult</a> of <font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Olive">f</font>,<font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="partfun1.html#K1">*</a> <font color="Olive">f</font> ) )
 <b>and </b><br/><a NAME="E2:53_1_2"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> implies <font color="Olive">x</font> is    <a href="topgrp_1.html#M3" target="_self">Homeomorphism</a> of <font color="Maroon">T</font> ) &amp; ( <font color="Olive">x</font> is    <a href="topgrp_1.html#M3" target="_self">Homeomorphism</a> of <font color="Maroon">T</font> implies <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> ) &amp; ( for <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> holds  the <a href="group_1.html#U1">mult</a> of <font color="Maroon">B</font> <a href="binop_1.html#K1">.</a> <font color="Olive">f</font>,<font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="partfun1.html#K1">*</a> <font color="Olive">f</font> ) )
 ;<br/>

<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/>
<a NAME="E3:53_1_2"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] )
 
<b>by </b><i/>;<br/>
<a NAME="E4:53_1_2"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] )
 
<b>by </b><i/>;<br/>
<a NAME="E5:53_1_2"/><i><font color="Green">E55</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>
 
<b>from </b><i><a class="ref" href="xboole_0.html#S2">XBOOLE_0:sch 2</a>(, );<br/></i>
<a NAME="E6:53_1_2"/>
for <font color="Olive">c</font>, <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> &amp; <font color="Olive">d</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> holds <br/>the <a href="group_1.html#U1">mult</a> of <font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Olive">c</font>,<font color="Olive">d</font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">B</font> <a href="binop_1.html#K1">.</a> <font color="Olive">c</font>,<font color="Olive">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:53_1_2_1"/>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">c</font>, <font color="Maroon">g</font> = <font color="Maroon">d</font> as    <a href="topgrp_1.html#M3" target="_self">Homeomorphism</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<b>thus </b>the <a href="group_1.html#U1">mult</a> of <font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c</font>,<font color="Maroon">d</font> = 
<font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
<b>by </b><i/>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">B</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:53_1_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="binop_1.html#T1">BINOP_1:1</a></i>;<br/>


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