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<b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A</font> implies <font color="Olive" title="b1">x</font> is    <a href="topgrp_1.html#M3" title="TOPGRP_1:mode.3">Homeomorphism</a> of <font color="Maroon" title="c1">T</font> ) &amp; ( <font color="Olive" title="b1">x</font> is    <a href="topgrp_1.html#M3" title="TOPGRP_1:mode.3">Homeomorphism</a> of <font color="Maroon" title="c1">T</font> implies <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A</font> ) &amp; ( for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font> being   <a href="topgrp_1.html#M3" title="TOPGRP_1:mode.3">Homeomorphism</a> of <font color="Maroon" title="c1">T</font> holds  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">f</font>,<font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">f</font> ) ) ) &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">B</font> implies <font color="Olive" title="b1">x</font> is    <a href="topgrp_1.html#M3" title="TOPGRP_1:mode.3">Homeomorphism</a> of <font color="Maroon" title="c1">T</font> ) &amp; ( <font color="Olive" title="b1">x</font> is    <a href="topgrp_1.html#M3" title="TOPGRP_1:mode.3">Homeomorphism</a> of <font color="Maroon" title="c1">T</font> implies <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">B</font> ) &amp; ( for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font> being   <a href="topgrp_1.html#M3" title="TOPGRP_1:mode.3">Homeomorphism</a> of <font color="Maroon" title="c1">T</font> holds  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">B</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">f</font>,<font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">f</font> ) ) ) implies <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:51_1_2"/><i><font color="Green" title="E29">A5</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A</font> implies <font color="Olive" title="b1">x</font> is    <a href="topgrp_1.html#M3" title="TOPGRP_1:mode.3">Homeomorphism</a> of <font color="Maroon" title="c1">T</font> ) &amp; ( <font color="Olive" title="b1">x</font> is    <a href="topgrp_1.html#M3" title="TOPGRP_1:mode.3">Homeomorphism</a> of <font color="Maroon" title="c1">T</font> implies <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A</font> ) &amp; ( for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font> being   <a href="topgrp_1.html#M3" title="TOPGRP_1:mode.3">Homeomorphism</a> of <font color="Maroon" title="c1">T</font> holds  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">f</font>,<font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">f</font> ) )
 <b>and </b><br/><a NAME="E2:51_1_2"/><i><font color="Green" title="E30">A6</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">B</font> implies <font color="Olive" title="b1">x</font> is    <a href="topgrp_1.html#M3" title="TOPGRP_1:mode.3">Homeomorphism</a> of <font color="Maroon" title="c1">T</font> ) &amp; ( <font color="Olive" title="b1">x</font> is    <a href="topgrp_1.html#M3" title="TOPGRP_1:mode.3">Homeomorphism</a> of <font color="Maroon" title="c1">T</font> implies <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">B</font> ) &amp; ( for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font> being   <a href="topgrp_1.html#M3" title="TOPGRP_1:mode.3">Homeomorphism</a> of <font color="Maroon" title="c1">T</font> holds  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">B</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">f</font>,<font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">f</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font></span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1 is    <a href="topgrp_1.html#M3" title="TOPGRP_1:mode.3">Homeomorphism</a> of <font color="Maroon" title="c1">T</font>;<br/>
<a NAME="E3:51_1_2"/><i><font color="Green" title="E31">A7</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] )
 
<b>by </b><i><a class="txt" href="#E1:51_1_2"><i><font color="Green" title="E29">A5</font></i></a></i>;<br/>
<a NAME="E4:51_1_2"/><i><font color="Green" title="E32">A8</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">B</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] )
 
<b>by </b><i><a class="txt" href="#E2:51_1_2"><i><font color="Green" title="E30">A6</font></i></a></i>;<br/>
<a NAME="E5:51_1_2"/><i><font color="Green" title="E33">A9</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">B</font>
 
<b>from </b><i><a class="ref" href="xboole_0.html#S2" title="XBOOLE_0:sch.2">XBOOLE_0:sch 2</a>(<a class="txt" href="#E3:51_1_2"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="#E4:51_1_2"><i><font color="Green" title="E32">A8</font></i></a>);<br/></i>
<a NAME="E6:51_1_2"/>
for <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A</font> holds <br/>the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">B</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">d</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c5">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A</font> implies the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">B</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> )</span><br/>

<b>assume </b><a NAME="E1:51_1_2_1"/>
( <font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c5">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">B</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c7">g</font> = <font color="Maroon" title="c5">d</font> as    <a href="topgrp_1.html#M3" title="TOPGRP_1:mode.3">Homeomorphism</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="#E1:51_1_2"><i><font color="Green" title="E29">A5</font></i></a></i>;<br/>
<b>thus </b>the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> = 
<font color="Maroon" title="c7">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">f</font>
<b>by </b><i><a class="txt" href="#E1:51_1_2"><i><font color="Green" title="E29">A5</font></i></a></i>
<br/>.= 
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">B</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>
<b>by </b><i><a class="txt" href="#E2:51_1_2"><i><font color="Green" title="E30">A6</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:51_1_2"/>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="#E5:51_1_2"><i><font color="Green" title="E33">A9</font></i></a>, <a class="ref" href="binop_1.html#T1" title="BINOP_1:th.1">BINOP_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
