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<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <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">F</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">A</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:50"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">F</font> is <a href="binop_1.html#V3">idempotent</a> &amp; <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> )
 <b>and </b><br/><a NAME="E2:50"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">F</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</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>;<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>let </b><font color="Maroon">g</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">Y</font>,<font color="Maroon">A</font>;<br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:50_1"/><b>then </b>
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="setwiseo.html#K1" target="_self">{}.</a> <font color="Maroon">X</font> &amp; <font color="Maroon">f</font> <a href="setwiseo.html#K8" target="_self">.:</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="setwiseo.html#K1" target="_self">{}.</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T149">RELAT_1:149</a></i>;<br/><a NAME="E3:50_1"/><b>then </b>
( <font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="setwiseo.html#K8" target="_self">.:</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="txt" href="setwiseo.html#E17"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="setwiseo.html#T18" target="_self">Th18</a>, </i>;<br/><b>hence </b><a NAME="E4:50_1"/>
<font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="setwiseo.html#K8" target="_self">.:</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:50"/>
<font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="setwiseo.html#K8" target="_self">.:</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="setwiseo.html#K7" target="_self">$$</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="setwiseo.html#E17"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="setwiseo.html#T13" target="_self">Th13</a></i>;<br/>


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