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







<b>assume </b><b>that </b><br/><a NAME="E1:48"/><i><font color="Green" title="E31">A1</font></i>: 
( <font color="Maroon" title="c3">F</font> is <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> &amp; <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> )
 <b>and </b><br/><a NAME="E2:48"/><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="c5">B1</font>, <font color="Maroon" title="c6">B2</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">X</font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1"/><i><font color="Green" title="E33">A3</font></i>: 
( <font color="Maroon" title="c5">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c6">B2</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="E2:48_1"/><i><font color="Green" title="E34">A4</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c2">X</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_1"/>
( <font color="Maroon" title="c6">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c5">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span><b>by </b><i><a class="txt" href="setwiseo.html#E1:48_1"><i><font color="Green" title="E33">A3</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_1_1"/><i><font color="Green" title="E35">A5</font></i>: 
<font color="Maroon" title="c6">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B1</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Maroon" title="c6">B2</font></span>)</span>,<font color="Maroon" title="c4">f</font> = 
<font color="Maroon" title="c3">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c4">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c3">F</font></span>)</span>
<b>by </b><i><a class="txt" href="setwiseo.html#E2:48"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="setwiseo.html#T23" target="_self" title="SETWISEO:th.23">Th23</a></i>
<br/>.= 
<font color="Maroon" title="c3">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c4">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c6">B2</font>,<font color="Maroon" title="c4">f</font></span>)</span>
<b>by </b><i><a class="txt" href="setwiseo.html#E1:48"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="setwiseo.html#E2:48"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="setwiseo.html#E2:48_1"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="setwiseo.html#E1:48_1_1_1"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="setwiseo.html#T40" target="_self" title="SETWISEO:th.40">Th40</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_1_2"/><i><font color="Green" title="E35">A6</font></i>: 
<font color="Maroon" title="c5">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B1</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Maroon" title="c6">B2</font></span>)</span>,<font color="Maroon" title="c4">f</font> = 
<font color="Maroon" title="c3">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c3">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c6">B2</font>,<font color="Maroon" title="c4">f</font></span>)</span>
<b>by </b><i><a class="txt" href="setwiseo.html#E2:48"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="setwiseo.html#T23" target="_self" title="SETWISEO:th.23">Th23</a></i>
<br/>.= 
<font color="Maroon" title="c3">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c4">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c6">B2</font>,<font color="Maroon" title="c4">f</font></span>)</span>
<b>by </b><i><a class="txt" href="setwiseo.html#E1:48"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="setwiseo.html#E2:48"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="setwiseo.html#E2:48_1"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="setwiseo.html#E1:48_1_1_2"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="setwiseo.html#T40" target="_self" title="SETWISEO:th.40">Th40</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:48"/>
<font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B1</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Maroon" title="c6">B2</font></span>)</span>,<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c4">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c6">B2</font>,<font color="Maroon" title="c4">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="setwiseo.html#E1:48"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="setwiseo.html#T30" target="_self" title="SETWISEO:th.30">Th30</a></i>;<br/>


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