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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">H1</font>, <font color="Olive">H2</font> being   <a href="group_2.html#M1">Subgroup</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">H1</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">H2</font> holds <br/><font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">H1</font> <a href="group_2.html#K9">/\</a> <font color="Olive">H2</font>;<br/>
<a NAME="E1:81_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">S1</font>, <font color="Olive">S2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="group_3.html#K1">Subgroups</a> <font color="Maroon">G</font>  ex <font color="Olive">B</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="group_3.html#K1">Subgroups</a> <font color="Maroon">G</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">S1</font>,<font color="Olive">S2</font>,<font color="Olive">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">S1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="group_3.html#K1">Subgroups</a> <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">S2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="group_3.html#K1">Subgroups</a> <font color="Maroon">G</font>;<br/>



<b>reconsider </b><font color="Maroon">H1</font> = <font color="Maroon">S1</font>, <font color="Maroon">H2</font> = <font color="Maroon">S2</font> as    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="group_3.html#D1">GROUP_3:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">H1</font> <a href="group_2.html#K9">/\</a> <font color="Maroon">H2</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="group_3.html#K1">Subgroups</a> <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="group_3.html#D1">GROUP_3:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">C</font>
;<br/>

<b>thus </b><a NAME="E3:81_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">S1</font>,<font color="Maroon">S2</font>,<font color="Maroon">C</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:81_1_1"/>
 ex <font color="Olive">o</font> being  <a href="binop_1.html#NM2">BinOp</a> of  <a href="group_3.html#K1">Subgroups</a> <font color="Maroon">G</font> st <br/>for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="group_3.html#K1">Subgroups</a> <font color="Maroon">G</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">o</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">b</font>]
 
<b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>();<br/></i>
<b>hence </b><a NAME="E3:81_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="binop_1.html#NM2">BinOp</a> of  <a href="group_3.html#K1">Subgroups</a> <font color="Maroon">G</font> st <br/>for <font color="Olive">S1</font>, <font color="Olive">S2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="group_3.html#K1">Subgroups</a> <font color="Maroon">G</font><br/> for <font color="Olive">H1</font>, <font color="Olive">H2</font> being   <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>  st <font color="Olive">S1</font> <a href="hidden.html#R1">=</a> <font color="Olive">H1</font> &amp; <font color="Olive">S2</font> <a href="hidden.html#R1">=</a> <font color="Olive">H2</font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">S1</font>,<font color="Olive">S2</font> <a href="hidden.html#R1">=</a> <font color="Olive">H1</font> <a href="group_2.html#K9">/\</a> <font color="Olive">H2</font>
 ;<br/>


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