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<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</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="c3">F</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">Y</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:49"/><i><font color="Green" title="E31">A1</font></i>: 
( <font color="Maroon" title="c3">F</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; <font color="Maroon" title="c3">F</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp; <font color="Maroon" title="c3">F</font> is <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> )
 <b>and </b><br/><a NAME="E2:49"/><i><font color="Green" title="E32">A2</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>;<br/><b>let </b><font color="Maroon" title="c6">A</font>, <font color="Maroon" title="c7">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/>



<b>assume </b><a NAME="E3:49"/><i><font color="Green" title="E33">A3</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c7">B</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1"/>
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E2:49_1"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T149" title="RELAT_1:th.149">RELAT_1:149</a></i>;<br/><a NAME="E3:49_1"/><b>then </b>
( <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c1">X</font> )
 <b>by </b><i><a class="txt" href="setwiseo.html#E3:49"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="setwiseo.html#T19" target="_self" title="SETWISEO:th.19">Th19</a></i>;<br/><a NAME="E4:49_1"/><b>then </b>
( <font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c5">g</font> <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> <font color="Maroon" title="c3">F</font> )
 <b>by </b><i><a class="txt" href="setwiseo.html#E1:49"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="setwiseo.html#E2:49"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="setwiseo.html#T40" target="_self" title="SETWISEO:th.40">Th40</a></i>;<br/><b>hence </b><a NAME="E5:49_1"/>
<font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c5">g</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:49"/>
<font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c5">g</font>
 <b>by </b><i><a class="txt" href="setwiseo.html#E1:49"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="setwiseo.html#E3:49"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="setwiseo.html#T35" target="_self" title="SETWISEO:th.35">Th35</a></i>;<br/>


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