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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font>,   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> $2;<br/>
<a NAME="E1:11_1_1"/><i><font color="Green" title="E10">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font>  ex <font color="Olive" title="b3">m</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">m</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">m</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Olive" title="b1">m</font>]</span><br/>

<b>reconsider </b><font color="Maroon" title="c4">xx</font> = <font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c5">yy</font> = <font color="Maroon" title="c3">y</font> as   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="autgroup.html#D1" target="_self" title="AUTGROUP:def.1">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">m</font> = <font color="Maroon" title="c4">xx</font> <a href="group_6.html#K7" title="GROUP_6:func.7">*</a> <font color="Maroon" title="c5">yy</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="autgroup.html#T8" target="_self" title="AUTGROUP:th.8">Th8</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">m</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c6">m</font>]</span><br/>

<b>thus </b><a NAME="E3:11_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c6">m</font>]
 ; <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>thus </b><a NAME="E2:11_1_1"/>
 ex <font color="Olive" title="b1">M</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of  <a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font> st <br/>for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>,<font color="Olive" title="b1">M</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="autgroup.html#E1:11_1_1"><i><font color="Green" title="E10">A1</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/>


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