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

<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">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="c4">F</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">A</font>;<br/>



<b>assume </b><a NAME="E1:44"/><i><font color="Green" title="E28">A1</font></i>: 
( <font color="Maroon" title="c4">F</font> is <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; <font color="Maroon" title="c4">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">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>assume </b><a NAME="E2:44"/><i><font color="Green" title="E29">A2</font></i>: 
<font color="Maroon" title="c5">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/>

<b>let </b><font color="Maroon" title="c6">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="c3">Y</font>;<br/><b>let </b><font color="Maroon" title="c7">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c1">A</font>;<br/>



<a NAME="E3:44"/><i><font color="Green" title="E30">A3</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<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><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> $1</span>)</span>,<font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> $1,<span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">f</font></span>)</span>;<br/>
<a NAME="E4:44"/><i><font color="Green" title="E31">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> holds  <font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<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/>

<a NAME="E1:44_1"/>
<font color="Maroon" title="c6">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <span class="p1"><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c8">x</font>
 
;<br/>
<b>hence </b><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c8">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<font color="Maroon" title="c7">g</font> = 
<font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<font color="Maroon" title="c7">g</font>
<b>by </b><i><a class="txt" href="setwiseo.html#E3:44"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="funct_1.html#T117" title="FUNCT_1:th.117">FUNCT_1:117</a></i>
<br/>.= 
<font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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#E1:44"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="setwiseo.html#T26" target="_self" title="SETWISEO:th.26">Th26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><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 class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">f</font></span>)</span>
<b>by </b><i><a class="txt" href="setwiseo.html#E1:44"><i><font color="Green" title="E28">A1</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>
<a NAME="E5:44"/><i><font color="Green" title="E32">A5</font></i>: 
for <font color="Olive" title="b1">B1</font>, <font color="Olive" title="b2">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>  st <font color="Olive" title="b1">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>  &amp; <font color="Olive" title="b2">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>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B2</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B1</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Olive" title="b2">B2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">B1</font>, <font color="Maroon" title="c9">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="E1:44_2"/><i><font color="Green" title="E33">A6</font></i>: 
( <font color="Maroon" title="c8">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>  &amp; <font color="Maroon" title="c9">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>  )
 ;<br/>

<a NAME="E2:44_2"/>
( <font color="Maroon" title="c8">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c9">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/>
<a NAME="E3:44_2"/><b>then </b>
( <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">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>  &amp; <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c9">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>  )
 
<b>by </b><i><a class="txt" href="setwiseo.html#E1:44_2"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E4:44_2"/><b>then </b>
( <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c8">B1</font> &amp; <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c9">B2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E5:44_2"/><b>then </b><i><font color="Green" title="E34">A7</font></i>: 
( <font color="Maroon" title="c6">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">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>  &amp; <font color="Maroon" title="c6">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c9">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>  )
 
<b>by </b><i><a class="txt" href="setwiseo.html#E3:44"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="relat_1.html#T151" title="RELAT_1:th.151">RELAT_1:151</a></i>;<br/>
<b>assume </b><a NAME="E6:44_2"/><i><font color="Green" title="E35">A8</font></i>: 
( <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">B1</font></span>)</span>,<font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c8">B1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">f</font></span>)</span> &amp; <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c9">B2</font></span>)</span>,<font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c9">B2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">f</font></span>)</span> )
 ;<br/>

<b>thus </b><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">B1</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Maroon" title="c9">B2</font></span>)</span></span>)</span>,<font color="Maroon" title="c7">g</font> = 
<font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">B1</font></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c9">B2</font></span>)</span></span>)</span>,<font color="Maroon" title="c7">g</font>
<b>by </b><i><a class="ref" href="relat_1.html#T153" title="RELAT_1:th.153">RELAT_1:153</a></i>
<br/>.= 
<font color="Maroon" title="c4">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">B1</font></span>)</span>,<font color="Maroon" title="c7">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c9">B2</font></span>)</span>,<font color="Maroon" title="c7">g</font></span>)</span>
<b>by </b><i><a class="txt" href="setwiseo.html#E1:44"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="setwiseo.html#E5:44_2"><i><font color="Green" title="E34">A7</font></i></a>, <a class="ref" href="setwiseo.html#T30" target="_self" title="SETWISEO:th.30">Th30</a></i>
<br/>.= 
<font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B1</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Maroon" title="c9">B2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">f</font></span>)</span>
<b>by </b><i><a class="txt" href="setwiseo.html#E1:44"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="setwiseo.html#E1:44_2"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="setwiseo.html#E6:44_2"><i><font color="Green" title="E35">A8</font></i></a>, <a class="ref" href="setwiseo.html#T30" target="_self" title="SETWISEO:th.30">Th30</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E6:44"/>
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>  st <font color="Olive" title="b1">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>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>]
 
<b>from </b><i><a class="ref" href="setwiseo.html#S3" target="_self" title="SETWISEO:sch.3">SETWISEO:sch 3</a>(<a class="txt" href="setwiseo.html#E4:44"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="setwiseo.html#E5:44"><i><font color="Green" title="E32">A5</font></i></a>);<br/></i>
<b>hence </b><a NAME="E7:44"/>
<font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c5">B</font></span>)</span>,<font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c5">B</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="setwiseo.html#E2:44"><i><font color="Green" title="E29">A2</font></i></a></i>;<br/>


</div>
