<?xml version="1.0"?>
<div class="add">

<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><a NAME="E1:35"/><i><font color="Green" title="E24">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> )
 ;<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/>

<b>assume </b><a NAME="E2:35"/><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c5">B1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>] <b>means </b>( $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B1</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> $1</span>)</span>,<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</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> $1,<font color="Maroon" title="c4">f</font></span>)</span> );<br/>
<a NAME="E3:35"/><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c2">X</font>]
 
;<br/>
<a NAME="E4:35"/><i><font color="Green" title="E27">A4</font></i>: 
for <font color="Olive" title="b1">B'</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">X</font><br/> for <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B'</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B'</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b2">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>let </b><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="c2">X</font>;<br/><b>let </b><font color="Maroon" title="c8">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:35_1"/><i><font color="Green" title="E28">A5</font></i>: 
( <font color="Maroon" title="c7">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B1</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Maroon" title="c7">B</font></span>)</span>,<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</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="c7">B</font>,<font color="Maroon" title="c4">f</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:35_1"/>
<font color="Maroon" title="c7">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_1"/>
( <font color="Maroon" title="c7">B</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="c7">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1"><b>suppose </b></a><a NAME="E1:35_1_1_1"/><i><font color="Green" title="E29">A6</font></i>: 
<font color="Maroon" title="c7">B</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">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B1</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c8">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c4">f</font> = 
<font color="Maroon" title="c3">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</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="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
<b>by </b><i><a class="txt" href="setwiseo.html#E1:35"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="setwiseo.html#E2:35"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="setwiseo.html#T29" target="_self" title="SETWISEO:th.29">Th29</a></i>
<br/>.= 
<font color="Maroon" title="c3">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</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> <span class="p2">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c8">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<font color="Maroon" title="c4">f</font></span>)</span>
<b>by </b><i><a class="txt" href="setwiseo.html#E1:35"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="setwiseo.html#E1:35_1_1_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="setwiseo.html#T26" target="_self" title="SETWISEO:th.26">Th26</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2"><b>suppose </b></a><a NAME="E1:35_1_1_2"/><i><font color="Green" title="E29">A7</font></i>: 
<font color="Maroon" title="c7">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>thus </b><font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B1</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c8">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c4">f</font> = 
<font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">B1</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Maroon" title="c7">B</font></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c8">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<font color="Maroon" title="c4">f</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>
<br/>.= 
<font color="Maroon" title="c3">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c4">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c4">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
<b>by </b><i><a class="txt" href="setwiseo.html#E1:35"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="setwiseo.html#E1:35_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="setwiseo.html#E1:35_1_1_2"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="setwiseo.html#T29" target="_self" title="SETWISEO:th.29">Th29</a>, <a class="ref" href="xboole_1.html#T15" title="XBOOLE_1:th.15">XBOOLE_1:15</a></i>
<br/>.= 
<font color="Maroon" title="c3">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c5">B1</font>,<font color="Maroon" title="c4">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c7">B</font>,<font color="Maroon" title="c4">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="setwiseo.html#E1:35"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="binop_1.html#D3" title="BINOP_1:def.3">BINOP_1:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c3">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</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> <span class="p2">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c8">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<font color="Maroon" title="c4">f</font></span>)</span>
<b>by </b><i><a class="txt" href="setwiseo.html#E1:35"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="setwiseo.html#E1:35_1_1_2"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="setwiseo.html#T29" target="_self" title="SETWISEO:th.29">Th29</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E5:35"/>
for <font color="Olive" title="b1">B2</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">X</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B2</font>]
 
<b>from </b><i><a class="ref" href="setwiseo.html#S4" target="_self" title="SETWISEO:sch.4">SETWISEO:sch 4</a>(<a class="txt" href="setwiseo.html#E3:35"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="setwiseo.html#E4:35"><i><font color="Green" title="E27">A4</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:35"/>
(  not <font color="Maroon" title="c6">B2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B1</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Maroon" title="c6">B2</font></span>)</span>,<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</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> )
 ;<br/>


</div>
