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<b>let </b><font color="Maroon">o1</font> be   <a href="binop_1.html#NM2">BinOp</a> of  <a href="group_6.html#K4" target="_self">Cosets</a> <font color="Maroon">N</font>, <font color="Maroon">o2</font> be   <a href="binop_1.html#NM2">BinOp</a> of  <a href="group_6.html#K4" target="_self">Cosets</a> <font color="Maroon">N</font>;<br/>

<b>assume </b><a NAME="E1:25_1_2"/><i><font color="Green">E46</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><br/> 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="Olive">W1</font> <a href="hidden.html#R1">=</a> <font color="Olive">A1</font> &amp; <font color="Olive">W2</font> <a href="hidden.html#R1">=</a> <font color="Olive">A2</font> holds <br/><font color="Maroon">o1</font> <a href="binop_1.html#K2">.</a> <font color="Olive">W1</font>,<font color="Olive">W2</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/>

<b>assume </b><a NAME="E2:25_1_2"/><i><font color="Green">E51</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><br/> 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="Olive">W1</font> <a href="hidden.html#R1">=</a> <font color="Olive">A1</font> &amp; <font color="Olive">W2</font> <a href="hidden.html#R1">=</a> <font color="Olive">A2</font> holds <br/><font color="Maroon">o2</font> <a href="binop_1.html#K2">.</a> <font color="Olive">W1</font>,<font color="Olive">W2</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/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:25_1_2_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_6.html#K4" target="_self">Cosets</a> <font color="Maroon">N</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="group_6.html#K4" target="_self">Cosets</a> <font color="Maroon">N</font> )
 ;<br/><b>then reconsider </b><font color="Maroon">W</font> = <font color="Maroon">x</font>, <font color="Maroon">V</font> = <font color="Maroon">y</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> ;<br/><b>reconsider </b><font color="Maroon">A1</font> = <font color="Maroon">x</font>, <font color="Maroon">A2</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/><a NAME="E4:25_1_2_1"/>
( <font color="Maroon">o1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">W</font>,<font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="group_2.html#K2">*</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">o2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">W</font>,<font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="group_2.html#K2">*</a> <font color="Maroon">A2</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E5:25_1_2_1"/><b>then </b>
( <font color="Maroon">o1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">o1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">o2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 ;<br/><b>hence </b><a NAME="E6:25_1_2_1"/>
<font color="Maroon">o1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:25_1_2"/>
<font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T1">BINOP_1:1</a></i>;<br/>


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