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

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">R</font> be   <a href="fuzzy_3.html#NM1">RMembership_Func</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">Q</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>







<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>] <b>means </b>verum;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> = <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">a<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p2">(<span class="default"><a href="card_3.html#K5">pi</a> <font color="Maroon">Q</font>,<span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="lfuzzy_0.html#K1">RealPoset</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> =  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">a<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5">pi</a> <font color="Maroon">Q</font>,<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> </span>}</span> ,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span>;<br/>
<a NAME="E1:53"/>
for <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds  <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p2">(<span class="default"><a href="card_3.html#K5">pi</a> <font color="Maroon">Q</font>,<span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="lfuzzy_0.html#K1">RealPoset</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5">pi</a> <font color="Maroon">Q</font>,<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> </span>}</span> ,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>

 <a href="lfuzzy_0.html#K4">FuzzyLattice</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span> = 
<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="yellow_1.html#K6">|^</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="lfuzzy_0.html#D4">LFUZZY_0:def 4</a></i>
<br/>.= 
 <a href="yellow_1.html#K5">product</a> <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="lfuzzy_0.html#K1">RealPoset</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_1.html#D5">YELLOW_1:def 5</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">Q</font> = <font color="Maroon">Q</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="lfuzzy_0.html#K1">RealPoset</a> <span class="p4"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span></span>)</span></span>)</span> ;<br/>
<a NAME="E3:53_1"/>
 <a href="waybel_3.html#K5">pi</a> <font color="Maroon">Q</font>,<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>hence </b><a NAME="E4:53_1"/>
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p2">(<span class="default"><a href="card_3.html#K5">pi</a> <font color="Maroon">Q</font>,<span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="lfuzzy_0.html#K1">RealPoset</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5">pi</a> <font color="Maroon">Q</font>,<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> </span>}</span> ,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:53"/><b>then </b><i><font color="Green">E49</font></i>: 
for <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">y</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">y</font>)
 
;<br/>
<a NAME="E3:53"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">y</font>) where <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">y</font>) where <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S5">FRAENKEL:sch 5</a>();<br/></i>
<b>hence </b><a NAME="E4:53"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p4">(<span class="default"><a href="card_3.html#K5">pi</a> <font color="Maroon">Q</font>,<span class="p5"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p4">(<span class="default"><a href="lfuzzy_0.html#K1">RealPoset</a> <span class="p5"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span></span>)</span></span>)</span> where <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> : verum </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K7">.</a> <span class="p0"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Olive">y'</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5">pi</a> <font color="Maroon">Q</font>,<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y'</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> </span>}</span> ,<span class="p3">(<span class="default"><a href="lfuzzy_0.html#K1">RealPoset</a> <span class="p4"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span></span>)</span> where <font color="Olive">y'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> : verum </span>}</span> 
 ;<br/>


</div>
