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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="group_6.html#K4" target="_self">Cosets</a> <font color="Maroon">N</font>,   <a href="subset_1.html#M1">Element</a> of  <a href="group_6.html#K4" target="_self">Cosets</a> <font color="Maroon">N</font>,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">A1</font>, <font color="Olive">A2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">A1</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">A2</font> holds <br/><font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">A1</font> <a href="group_2.html#K2">*</a> <font color="Olive">A2</font>;<br/>
<a NAME="E1:25_1_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">W1</font>, <font color="Olive">W2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="group_6.html#K4" target="_self">Cosets</a> <font color="Maroon">N</font>  ex <font color="Olive">V</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="group_6.html#K4" target="_self">Cosets</a> <font color="Maroon">N</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">W1</font>,<font color="Olive">W2</font>,<font color="Olive">V</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">W1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="group_6.html#K4" target="_self">Cosets</a> <font color="Maroon">N</font>, <font color="Maroon">W2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="group_6.html#K4" target="_self">Cosets</a> <font color="Maroon">N</font>;<br/>

<b>reconsider </b><font color="Maroon">A1</font> = <font color="Maroon">W1</font>, <font color="Maroon">A2</font> = <font color="Maroon">W2</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">A1</font> <a href="group_2.html#K2">*</a> <font color="Maroon">A2</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="group_6.html#K4" target="_self">Cosets</a> <font color="Maroon">N</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">C</font>
;<br/>

<b>thus </b><a NAME="E3:25_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">W1</font>,<font color="Maroon">W2</font>,<font color="Maroon">C</font>]
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E2:25_1_1"/>
 ex <font color="Olive">B</font> being  <a href="binop_1.html#NM2">BinOp</a> of  <a href="group_6.html#K4" target="_self">Cosets</a> <font color="Maroon">N</font> st <br/>for <font color="Olive">W1</font>, <font color="Olive">W2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="group_6.html#K4" target="_self">Cosets</a> <font color="Maroon">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">W1</font>,<font color="Olive">W2</font>,<font color="Olive">B</font> <a href="binop_1.html#K2">.</a> <font color="Olive">W1</font>,<font color="Olive">W2</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>();</i><br/>


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