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<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">o</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/>





<b>assume </b><a NAME="E1:81"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">a</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">o</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="monoid_1.html#K21" target="_self">.:^2</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">x</font> = 
<font color="Maroon">o</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">x</font></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">D</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">x</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="monoid_1.html#K21" target="_self">.:^2</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> = 
<font color="Maroon">o</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">x</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">D</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">x</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:81"/><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">o</font> <a href="monoid_1.html#K21" target="_self">.:^2</a> 
 <b>by </b><i><a class="ref" href="binop_1.html#T11">BINOP_1:11</a></i>;<br/>

<b>hence </b><a NAME="E4:81"/>
 ex <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> st <font color="Olive">A</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">o</font> <a href="monoid_1.html#K21" target="_self">.:^2</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="setwiseo.html#D2">SETWISEO:def 2</a><br/>

<b>thus </b><a NAME="E5:81"/>
 <a href="binop_1.html#K3">the_unity_wrt</a> <span class="p1">(<span class="default"><font color="Maroon">o</font> <a href="monoid_1.html#K21" target="_self">.:^2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="binop_1.html#D8">BINOP_1:def 8</a></i>;<br/>


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