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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">Y</font>, <font color="Maroon" title="c3">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>, <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c2">Y</font>;<br/><b>let </b><font color="Maroon" title="c5">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c2">Y</font>, <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c3">Z</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:77"/><i><font color="Green" title="E49">A1</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c3">Z</font>
 <b>and </b><br/><a NAME="E2:77"/><i><font color="Green" title="E50">A2</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c2">Y</font> holds  <font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">B</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">X</font>;<br/>

<a NAME="E3:77"/><i><font color="Green" title="E51">A3</font></i>: 
(  <a href="setwiseo.html#K9" title="SETWISEO:func.9">FinUnion</a> <font color="Maroon" title="c3">Z</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp;  <a href="setwiseo.html#K9" title="SETWISEO:func.9">FinUnion</a> <font color="Maroon" title="c3">Z</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp;  <a href="setwiseo.html#K9" title="SETWISEO:func.9">FinUnion</a> <font color="Maroon" title="c3">Z</font> is <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> &amp;  <a href="setwiseo.html#K9" title="SETWISEO:func.9">FinUnion</a> <font color="Maroon" title="c3">Z</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a>  )
 
<b>by </b><i><a class="ref" href="setwiseo.html#T49" target="_self" title="SETWISEO:th.49">Th49</a>, <a class="ref" href="setwiseo.html#T50" target="_self" title="SETWISEO:th.50">Th50</a>, <a class="ref" href="setwiseo.html#T51" target="_self" title="SETWISEO:th.51">Th51</a>, <a class="ref" href="setwiseo.html#T53" target="_self" title="SETWISEO:th.53">Th53</a></i>;<br/>
<a NAME="E4:77"/><i><font color="Green" title="E52">A4</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K9" title="SETWISEO:func.9">FinUnion</a> <font color="Maroon" title="c3">Z</font></span>)</span>
 
<b>by </b><i><a class="txt" href="setwiseo.html#E1:77"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="setwiseo.html#T55" target="_self" title="SETWISEO:th.55">Th55</a></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" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c2">Y</font>;<br/><b>thus </b><font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Maroon" title="c8">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font></span>)</span>
<b>by </b><i><a class="txt" href="setwiseo.html#E2:77"><i><font color="Green" title="E50">A2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setwiseo.html#K9" title="SETWISEO:func.9">FinUnion</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font></span>)</span>
<b>by </b><i><a class="ref" href="setwiseo.html#D4" target="_self" title="SETWISEO:def.4">Def4</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:77"/>
<font color="Maroon" title="c5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <font color="Maroon" title="c6">B</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="setwiseo.html#E3:77"><i><font color="Green" title="E51">A3</font></i></a>, <a class="txt" href="setwiseo.html#E4:77"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="setwiseo.html#T65" target="_self" title="SETWISEO:th.65">Th65</a></i>;<br/>


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