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<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">Y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>, <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">Y</font>, <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">Z</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:77"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1" target="_self">{}.</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwiseo.html#K1" target="_self">{}.</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E2:77"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">Y</font> holds  <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="setwiseo.html#K5" target="_self">\/</a> <font color="Olive">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="setwiseo.html#K5" target="_self">\/</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span>
 ;<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/>

<a NAME="E3:77"/><i><font color="Green">E16</font></i>: 
(  <a href="setwiseo.html#K9" target="_self">FinUnion</a> <font color="Maroon">Z</font> is <a href="binop_1.html#V2">associative</a> &amp;  <a href="setwiseo.html#K9" target="_self">FinUnion</a> <font color="Maroon">Z</font> is <a href="binop_1.html#V1">commutative</a> &amp;  <a href="setwiseo.html#K9" target="_self">FinUnion</a> <font color="Maroon">Z</font> is <a href="binop_1.html#V3">idempotent</a> &amp;  <a href="setwiseo.html#K9" target="_self">FinUnion</a> <font color="Maroon">Z</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  )
 
<b>by </b><i><a class="ref" href="setwiseo.html#T18" target="_self">Th18</a>, <a class="ref" href="setwiseo.html#T19" target="_self">Th19</a>, , </i>;<br/>
<a NAME="E4:77"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1" target="_self">{}.</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K9" target="_self">FinUnion</a> <font color="Maroon">Z</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">Y</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">Y</font>;<br/><b>thus </b><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="setwiseo.html#K5" target="_self">\/</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="setwiseo.html#K5" target="_self">\/</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="setwiseo.html#K9" target="_self">FinUnion</a> <font color="Maroon">Z</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="txt" href="setwiseo.html#E17"><i><font color="Green">Lemma32</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:77"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K10" target="_self">FinUnion</a> <font color="Maroon">B</font>,<font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwiseo.html#K10" target="_self">FinUnion</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="ref" href="setwiseo.html#T23" target="_self">Th23</a></i>;<br/>


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