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<b>let </b><font color="Maroon">Y</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">X</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">F</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/>







<b>assume </b><a NAME="E1:37"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">F</font> is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">F</font> is <a href="binop_1.html#V2">associative</a> &amp; <font color="Maroon">F</font> is <a href="binop_1.html#V3">idempotent</a> )
 ;<br/>

<b>let </b><font color="Maroon">B</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>, <font color="Maroon">C</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:37"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E3:37"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E4:37"/>
<font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="setwiseo.html#T12" target="_self">Th12</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<b>hence </b><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">B</font>,<font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">C</font>,<font color="Maroon">f</font></span>)</span> = 
<font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="setwiseo.html#K5" target="_self">\/</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">f</font>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">C</font>,<font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="setwiseo.html#T12" target="_self">Th12</a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/><br/>


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