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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">e</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">G</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:80"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">F</font> is <a href="binop_1.html#V2">associative</a>
 <b>and </b><br/><a NAME="E2:80"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">F</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a> 
 <b>and </b><br/><a NAME="E3:80"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">F</font> <a href="finseqop.html#NR2" target="_self">has_an_inverseOp</a> 
 <b>and </b><br/><a NAME="E4:80"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">G</font> <a href="binop_1.html#R6">is_distributive_wrt</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E5:80"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">F</font>
 ;<br/>

<b>let </b><font color="Maroon">d</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/>

<b>set </b><font color="Maroon">ed</font> = <font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">e</font>,<font color="Maroon">d</font>;<br/>
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">e</font>,<font color="Maroon">d</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">e</font>,<font color="Maroon">d</font></span>)</span> = 
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">e</font>,<font color="Maroon">e</font></span>)</span>,<font color="Maroon">d</font>
<b>by </b><i>, <a class="ref" href="binop_1.html#T23">BINOP_1:23</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">e</font>,<font color="Maroon">d</font>
<b>by </b><i><a class="ref" href="finseqop.html#T8" target="_self">Th8</a>, , <a class="ref" href="setwiseo.html#T23">SETWISEO:23</a></i>
;<br/>
<b>hence </b><a NAME="E7:80"/>
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">e</font>,<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="finseqop.html#E4:80"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseqop.html#T8" target="_self">Th8</a>, , , <a class="txt" href="finseqop.html#E2:80"><i><font color="Green">E37</font></i></a></i>;<br/>

<b>set </b><font color="Maroon">de</font> = <font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d</font>,<font color="Maroon">e</font>;<br/>
<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d</font>,<font color="Maroon">e</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d</font>,<font color="Maroon">e</font></span>)</span> = 
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">e</font>,<font color="Maroon">e</font></span>)</span>
<b>by </b><i>, <a class="ref" href="binop_1.html#T23">BINOP_1:23</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d</font>,<font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="finseqop.html#T8" target="_self">Th8</a>, , <a class="ref" href="setwiseo.html#T23">SETWISEO:23</a></i>
;<br/>
<b>hence </b><a NAME="E9:80"/>
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">d</font>,<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="finseqop.html#E4:80"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseqop.html#T8" target="_self">Th8</a>, , , <a class="txt" href="finseqop.html#E2:80"><i><font color="Green">E37</font></i></a></i>;<br/>


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