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<b>let </b><font color="Maroon" title="c3">o1</font>, <font color="Maroon" title="c4">o2</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c2">N</font>;<br/>

<b>assume </b><a NAME="E1:26_1_2"/><i><font color="Green" title="E18">A2</font></i>: 
for <font color="Olive" title="b1">W1</font>, <font color="Olive" title="b2">W2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c2">N</font><br/> for <font color="Olive" title="b3">A1</font>, <font color="Olive" title="b4">A2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A1</font> &amp; <font color="Olive" title="b2">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">A2</font> holds <br/><font color="Maroon" title="c3">o1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">W1</font>,<font color="Olive" title="b2">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Olive" title="b4">A2</font>
 ;<br/>

<b>assume </b><a NAME="E2:26_1_2"/><i><font color="Green" title="E19">A3</font></i>: 
for <font color="Olive" title="b1">W1</font>, <font color="Olive" title="b2">W2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c2">N</font><br/> for <font color="Olive" title="b3">A1</font>, <font color="Olive" title="b4">A2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A1</font> &amp; <font color="Olive" title="b2">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">A2</font> holds <br/><font color="Maroon" title="c4">o2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">W1</font>,<font color="Olive" title="b2">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Olive" title="b4">A2</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:26_1_2_1"/><i><font color="Green" title="E20">A4</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c2">N</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c2">N</font> )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c7">W</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c8">V</font> = <font color="Maroon" title="c6">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c2">N</font> ;<br/><b>reconsider </b><font color="Maroon" title="c9">A1</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c10">A2</font> = <font color="Maroon" title="c6">y</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="group_6.html#E1:26_1_2_1"><i><font color="Green" title="E20">A4</font></i></a></i>;<br/><a NAME="E4:26_1_2_1"/>
( <font color="Maroon" title="c3">o1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c7">W</font>,<font color="Maroon" title="c8">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c10">A2</font> &amp; <font color="Maroon" title="c4">o2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c7">W</font>,<font color="Maroon" title="c8">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c10">A2</font> )
 <b>by </b><i><a class="txt" href="group_6.html#E1:26_1_2"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="group_6.html#E2:26_1_2"><i><font color="Green" title="E19">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E5:26_1_2_1"/>
<font color="Maroon" title="c3">o1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:26_1_2"/>
<font color="Maroon" title="c3">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o2</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T1" title="BINOP_1:th.1">BINOP_1:1</a></i>;<br/>


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