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

<b>let </b><font color="Maroon" title="c1">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="c2">R</font> be   <a href="fuzzy_3.html#NM1" title="FUZZY_3:NM.1">RMembership_Func</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>;<br/>





<b>set </b><font color="Maroon" title="c4">FL</font> =  <a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c5">RP</font> =  <a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>;<br/>
<a NAME="E1:61"/>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span> ,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span> ,<span class="p2">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p3"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="lfuzzy_0.html#D5" title="LFUZZY_0:def.5">LFUZZY_0:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">F</font> =  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span> ,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> as   <a href="fuzzy_3.html#NM1" title="FUZZY_3:NM.1">RMembership_Func</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E3:61"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c3">Q</font>,<span class="p4">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p5"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>

<a NAME="E1:61_1"/><i><font color="Green" title="E40">A1</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">t</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:61_1_1"/>
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span><b> such that </b><br/><a NAME="E3:61_1_1"/><i><font color="Green" title="E41">A2</font></i>: 
( <font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Maroon" title="c10">r</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c10">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> )
 ;<br/>
<a NAME="E4:61_1_1"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Maroon" title="c10">r</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="lfuzzy_0.html#K6" title="LFUZZY_0:func.6">@</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Maroon" title="c10">r</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font>
 
<b>by </b><i><a class="ref" href="lfuzzy_0.html#D6" title="LFUZZY_0:def.6">LFUZZY_0:def 6</a></i>;<br/>
<b>hence </b><a NAME="E5:61_1_1"/>
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="lfuzzy_1.html#E3:61_1_1"><i><font color="Green" title="E41">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:61_1"/><i><font color="Green" title="E41">A3</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <span class="p5">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c3">Q</font>,<span class="p0">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p4">(<span class="default"><a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <font color="Maroon" title="c3">Q</font>,<span class="p5"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p4">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p5"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2"><b>proof </b></a><div class="add">

<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 <font color="Maroon" title="c1">X</font>] <b>means </b>verum;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c3">Q</font>,<span class="p4">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p5"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,$1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">$1,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> = <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p2">(<span class="default"><a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <font color="Maroon" title="c3">Q</font>,<span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,$1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">$1,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>;<br/>
<a NAME="E1:61_1_2"/>
for <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c3">Q</font>,<span class="p4">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p5"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p2">(<span class="default"><a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <font color="Maroon" title="c3">Q</font>,<span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>

<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c3">Q</font>,<span class="p3">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p4"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c3">Q</font>,<span class="p2">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p3"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span></span>)</span> <a href="lfuzzy_0.html#K9" title="LFUZZY_0:func.9">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="ref" href="lfuzzy_0.html#D5" title="LFUZZY_0:def.5">LFUZZY_0:def 5</a></i>
<br/>.= 
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">(<span class="default"><a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <font color="Maroon" title="c3">Q</font>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="lfuzzy_1.html#T32" target="_self" title="LFUZZY_1:th.32">Th32</a></i>
;<br/>
<b>hence </b><a NAME="E2:61_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c3">Q</font>,<span class="p4">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p5"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p2">(<span class="default"><a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <font color="Maroon" title="c3">Q</font>,<span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:61_1_2"/><b>then </b><i><font color="Green" title="E42">A4</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">y</font>)
 
;<br/>
<b>thus </b><a NAME="E3:61_1_2"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">y</font>) where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">y'</font>) where <font color="Olive" title="b1">y'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y'</font>] </span>}</span> 
 <b>from </b><i><a class="ref" href="fraenkel.html#S5" title="FRAENKEL:sch.5">FRAENKEL:sch 5</a>(<a class="txt" href="lfuzzy_1.html#E2:61_1_2"><i><font color="Green" title="E42">A4</font></i></a>);</i><br/>


</div><b>end;</b></div>
<a NAME="E3:61_1"/><i><font color="Green" title="E42">A5</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p4">(<span class="default"><a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <font color="Maroon" title="c3">Q</font>,<span class="p5"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p4">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p5"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p0"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y'</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b2">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> : <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <font color="Maroon" title="c3">Q</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> </span>}</span> ,<span class="p3">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p4"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">y'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_3"><b>proof </b></a><div class="add">

<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 <font color="Maroon" title="c1">X</font>] <b>means </b>verum;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> = <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p2">(<span class="default"><a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <font color="Maroon" title="c3">Q</font>,<span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,$1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">$1,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> =  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">$1,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <font color="Maroon" title="c3">Q</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,$1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> </span>}</span> ,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>;<br/>
<a NAME="E1:61_1_3"/>
for <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p2">(<span class="default"><a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <font color="Maroon" title="c3">Q</font>,<span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b2">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> : <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <font color="Maroon" title="c3">Q</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> </span>}</span> ,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>

 <a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> = 
<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="ref" href="lfuzzy_0.html#D4" title="LFUZZY_0:def.4">LFUZZY_0:def 4</a></i>
<br/>.= 
 <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_1.html#D5" title="YELLOW_1:def.5">YELLOW_1:def 5</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">Q</font> = <font color="Maroon" title="c3">Q</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default"><span class="p3"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p4"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span></span>)</span> ;<br/>
<a NAME="E3:61_1_3_1"/>
 <a href="waybel_3.html#K5" title="WAYBEL_3:func.5">pi</a> <font color="Maroon" title="c10">Q</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>hence </b><a NAME="E4:61_1_3_1"/>
<span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p2">(<span class="default"><a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <font color="Maroon" title="c3">Q</font>,<span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <font color="Maroon" title="c3">Q</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> </span>}</span> ,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="lfuzzy_0.html#T15" title="LFUZZY_0:th.15">LFUZZY_0:15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:61_1_3"/><b>then </b><i><font color="Green" title="E43">A6</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">y</font>)
 
;<br/>
<a NAME="E3:61_1_3"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">y</font>) where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">y'</font>) where <font color="Olive" title="b1">y'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y'</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S5" title="FRAENKEL:sch.5">FRAENKEL:sch 5</a>(<a class="txt" href="lfuzzy_1.html#E2:61_1_3"><i><font color="Green" title="E43">A6</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:61_1_3"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p4">(<span class="default"><a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <font color="Maroon" title="c3">Q</font>,<span class="p5"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p4">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p5"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p0"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y'</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b2">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> : <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <font color="Maroon" title="c3">Q</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> </span>}</span> ,<span class="p3">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p4"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">y'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:61_1"/><i><font color="Green" title="E43">A7</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p0"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b2">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> : <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <font color="Maroon" title="c3">Q</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> </span>}</span> ,<span class="p3">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p4"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b2">r</font> <a href="lfuzzy_0.html#K9" title="LFUZZY_0:func.9">.</a> <span class="p0"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p0"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y'</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b2">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span> ,<span class="p3">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p4"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">y'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_4"><b>proof </b></a><div class="add">

<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 <font color="Maroon" title="c1">X</font>] <b>means </b>verum;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> =  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">$1,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <font color="Maroon" title="c3">Q</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,$1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> </span>}</span> ,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> =  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">r</font> <a href="lfuzzy_0.html#K9" title="LFUZZY_0:func.9">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,$1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">$1,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span> ,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>;<br/>
<a NAME="E1:61_1_4"/>
for <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds   <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b2">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> : <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <font color="Maroon" title="c3">Q</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> </span>}</span> ,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b2">r</font> <a href="lfuzzy_0.html#K9" title="LFUZZY_0:func.9">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b2">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span> ,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>set </b><font color="Maroon" title="c10">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <font color="Maroon" title="c3">Q</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c11">B</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">r</font> <a href="lfuzzy_0.html#K9" title="LFUZZY_0:func.9">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span> ;<br/>
<a NAME="E1:61_1_4_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <font color="Maroon" title="c3">Q</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">r</font> <a href="lfuzzy_0.html#K9" title="LFUZZY_0:func.9">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_4_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:61_1_4_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <font color="Maroon" title="c3">Q</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">r</font> <a href="lfuzzy_0.html#K9" title="LFUZZY_0:func.9">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_4_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:61_1_4_1_1_1"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <font color="Maroon" title="c3">Q</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span><b> such that </b><br/><a NAME="E3:61_1_4_1_1_1"/><i><font color="Green" title="E44">A8</font></i>: 
( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> &amp; <font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <font color="Maroon" title="c3">Q</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c14">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:61_1_4_1_1_1"/><i><font color="Green" title="E45">A9</font></i>: 
( <font color="Maroon" title="c14">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> &amp; <font color="Maroon" title="c13">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 <b>by </b><i><a class="txt" href="lfuzzy_1.html#E3:61_1_4_1_1_1"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="card_3.html#D6" title="CARD_3:def.6">CARD_3:def 6</a></i>;<br/>
<b>thus </b><a NAME="E6:61_1_4_1_1_1"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">r</font> <a href="lfuzzy_0.html#K9" title="LFUZZY_0:func.9">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="lfuzzy_1.html#E3:61_1_4_1_1_1"><i><font color="Green" title="E44">A8</font></i></a>, <a class="txt" href="lfuzzy_1.html#E5:61_1_4_1_1_1"><i><font color="Green" title="E45">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:61_1_4_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">r</font> <a href="lfuzzy_0.html#K9" title="LFUZZY_0:func.9">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <font color="Maroon" title="c3">Q</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_4_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:61_1_4_1_1_2"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">r</font> <a href="lfuzzy_0.html#K9" title="LFUZZY_0:func.9">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span><b> such that </b><br/><a NAME="E3:61_1_4_1_1_2"/><i><font color="Green" title="E44">A10</font></i>: 
( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">r</font> <a href="lfuzzy_0.html#K9" title="LFUZZY_0:func.9">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> &amp; <font color="Maroon" title="c13">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> )
 ;<br/>
<a NAME="E4:61_1_4_1_1_2"/>
<font color="Maroon" title="c13">r</font> <a href="lfuzzy_0.html#K9" title="LFUZZY_0:func.9">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <font color="Maroon" title="c3">Q</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E3:61_1_4_1_1_2"><i><font color="Green" title="E44">A10</font></i></a>, <a class="ref" href="card_3.html#D6" title="CARD_3:def.6">CARD_3:def 6</a></i>;<br/>
<b>hence </b><a NAME="E5:61_1_4_1_1_2"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <font color="Maroon" title="c3">Q</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> </span>}</span> 
 <b>by </b><i><a class="txt" href="lfuzzy_1.html#E3:61_1_4_1_1_2"><i><font color="Green" title="E44">A10</font></i></a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:61_1_4_1"/>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">b</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <font color="Maroon" title="c3">Q</font>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> </span>}</span> ,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">r</font> <a href="lfuzzy_0.html#K9" title="LFUZZY_0:func.9">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span> ,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:61_1_4"/><b>then </b><i><font color="Green" title="E44">A11</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">y</font>)
 
;<br/>
<b>thus </b><a NAME="E3:61_1_4"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">y</font>) where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">y</font>) where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>] </span>}</span> 
 <b>from </b><i><a class="ref" href="fraenkel.html#S5" title="FRAENKEL:sch.5">FRAENKEL:sch 5</a>(<a class="txt" href="lfuzzy_1.html#E2:61_1_4"><i><font color="Green" title="E44">A11</font></i></a>);</i><br/>


</div><b>end;</b></div>
<a NAME="E5:61_1"/><i><font color="Green" title="E44">A12</font></i>: 
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b2">r</font> <a href="lfuzzy_0.html#K9" title="LFUZZY_0:func.9">.</a> <span class="p0"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p0"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b2">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span> ,<span class="p3">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p4"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</span> ,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b1">r'</font> <a href="lfuzzy_0.html#K9" title="LFUZZY_0:func.9">.</a> <span class="p0"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p0"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</span> ,<span class="p3">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p4"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">r'</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span> ,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_5"><b>proof </b></a><div class="add">

<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 <font color="Maroon" title="c1">X</font>] <b>means </b>verum;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>] <b>means </b>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>,  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> = <span class="p1">(<span class="default">$2 <a href="lfuzzy_0.html#K9" title="LFUZZY_0:func.9">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,$1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">$1,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>,  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> = <span class="p1">(<span class="default">$2 <a href="lfuzzy_0.html#K9" title="LFUZZY_0:func.9">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,$1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">$1,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>;<br/>
<a NAME="E1:61_1_5"/><i><font color="Green" title="E45">A13</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">r</font>] holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">r</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">r</font>)
 
;<br/>
<b>thus </b><a NAME="E2:61_1_5"/>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p3">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">r</font>) where <font color="Olive" title="b2">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">r</font>] </span>}</span> ,<span class="p3">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p4"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>] </span>}</span> ,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p3">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">y'</font>,<font color="Olive" title="b1">r'</font>) where <font color="Olive" title="b2">y'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">y'</font>] </span>}</span> ,<span class="p3">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p4"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">r'</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">r'</font>] </span>}</span> ,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>
 <b>from </b><i><a class="ref" href="lfuzzy_0.html#S5" title="LFUZZY_0:sch.5">LFUZZY_0:sch 5</a>(<a class="txt" href="lfuzzy_1.html#E1:61_1_5"><i><font color="Green" title="E45">A13</font></i></a>);</i><br/>


</div><b>end;</b></div>
<a NAME="E6:61_1"/><i><font color="Green" title="E45">A14</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b1">r</font> <a href="lfuzzy_0.html#K9" title="LFUZZY_0:func.9">.</a> <span class="p0"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p0"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</span> ,<span class="p3">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p4"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Olive" title="b1">r'</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p0"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p0"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</span> ,<span class="p3">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p4"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">r'</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_6"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>] <b>means </b>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> =  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default">$1 <a href="lfuzzy_0.html#K9" title="LFUZZY_0:func.9">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</span> ,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> =  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> $1</span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</span> ,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>;<br/>
<a NAME="E1:61_1_6"/>
for <font color="Olive" title="b1">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> holds <br/> <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">r</font> <a href="lfuzzy_0.html#K9" title="LFUZZY_0:func.9">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</span> ,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</span> ,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:61_1_6_1"/>
<font color="Maroon" title="c9">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font>
 ;<br/>

<a NAME="E2:61_1_6_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="lfuzzy_0.html#K9" title="LFUZZY_0:func.9">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_6_1_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>] <b>means </b>verum;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="lfuzzy_0.html#K9" title="LFUZZY_0:func.9">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,$1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">$1,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,$1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">$1,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>;<br/>
<a NAME="E1:61_1_6_1_1"/><i><font color="Green" title="E46">A15</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">y</font>)
 
<b>by </b><i><a class="ref" href="lfuzzy_0.html#D5" title="LFUZZY_0:def.5">LFUZZY_0:def 5</a></i>;<br/>
<b>thus </b><a NAME="E2:61_1_6_1_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">y</font>) where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">y</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">y</font>) where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">y</font>] </span>}</span> 
 <b>from </b><i><a class="ref" href="fraenkel.html#S5" title="FRAENKEL:sch.5">FRAENKEL:sch 5</a>(<a class="txt" href="lfuzzy_1.html#E1:61_1_6_1_1"><i><font color="Green" title="E46">A15</font></i></a>);</i><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:61_1_6_1"/>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="lfuzzy_0.html#K9" title="LFUZZY_0:func.9">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</span> ,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</span> ,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:61_1_6"/><b>then </b><i><font color="Green" title="E46">A16</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">r</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">r</font>)
 
;<br/>
<b>thus </b><a NAME="E3:61_1_6"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">r</font>) where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">r</font>) where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] </span>}</span> 
 <b>from </b><i><a class="ref" href="fraenkel.html#S6" title="FRAENKEL:sch.6">FRAENKEL:sch 6</a>(<a class="txt" href="lfuzzy_1.html#E2:61_1_6"><i><font color="Green" title="E46">A16</font></i></a>);</i><br/>


</div><b>end;</b></div>
<a NAME="E7:61_1"/><i><font color="Green" title="E46">A17</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p0"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p0"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</span> ,<span class="p3">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p4"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Olive" title="b1">r'</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> where <font color="Olive" title="b1">r'</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_7"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>] <b>means </b>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> =  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> $1</span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</span> ,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> $1</span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/>
<a NAME="E1:61_1_7"/><i><font color="Green" title="E47">A18</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">r</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">r</font>)
 
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E50"><i><font color="Green" title="E33">Lm6</font></i></a></i>;<br/>
<b>thus </b><a NAME="E2:61_1_7"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">r</font>) where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">r</font>) where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] </span>}</span> 
 <b>from </b><i><a class="ref" href="fraenkel.html#S6" title="FRAENKEL:sch.6">FRAENKEL:sch 6</a>(<a class="txt" href="lfuzzy_1.html#E1:61_1_7"><i><font color="Green" title="E47">A18</font></i></a>);</i><br/>


</div><b>end;</b></div>
<a NAME="E8:61_1"/><i><font color="Green" title="E47">A19</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span> ,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_8"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c9">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c10">B</font> =  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span> ,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/>
<b>thus </b><a NAME="E1:61_1_8"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span> ,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:61_1_8_1"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span><b> such that </b><br/><a NAME="E3:61_1_8_1"/><i><font color="Green" title="E48">A20</font></i>: 
( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Maroon" title="c12">r</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c12">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> )
 ;<br/>
<a NAME="E4:61_1_8_1"/>
<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Maroon" title="c12">r</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Olive" title="b1">r'</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> where <font color="Olive" title="b1">r'</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E3:61_1_8_1"><i><font color="Green" title="E48">A20</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:61_1_8_1"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span> ,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="lfuzzy_1.html#E3:61_1_8_1"><i><font color="Green" title="E48">A20</font></i></a>, <a class="ref" href="card_3.html#D6" title="CARD_3:def.6">CARD_3:def 6</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:61_1_8"/>
 <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span> ,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_8_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:61_1_8_2"/>
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span> ,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:61_1_8_2"/><i><font color="Green" title="E48">A21</font></i>: 
( <font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Olive" title="b1">r'</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> where <font color="Olive" title="b1">r'</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span>  &amp; <font color="Maroon" title="c11">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 <b>by </b><i><a class="ref" href="card_3.html#D6" title="CARD_3:def.6">CARD_3:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span><b> such that </b><br/><a NAME="E5:61_1_8_2"/><i><font color="Green" title="E49">A22</font></i>: 
( <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c13">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> )
 <b>by </b><i><a class="txt" href="lfuzzy_1.html#E3:61_1_8_2"><i><font color="Green" title="E48">A21</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:61_1_8_2"/>
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="lfuzzy_1.html#E3:61_1_8_2"><i><font color="Green" title="E48">A21</font></i></a>, <a class="txt" href="lfuzzy_1.html#E5:61_1_8_2"><i><font color="Green" title="E49">A22</font></i></a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c3">Q</font>,<span class="p4">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p5"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font> = 
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <span class="p5">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c3">Q</font>,<span class="p0">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c8">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</span> ,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E50"><i><font color="Green" title="E33">Lm6</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">F</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font>
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E1:61_1"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="lfuzzy_1.html#E2:61_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="lfuzzy_1.html#E3:61_1"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="lfuzzy_1.html#E4:61_1"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="lfuzzy_1.html#E5:61_1"><i><font color="Green" title="E44">A12</font></i></a>, <a class="txt" href="lfuzzy_1.html#E6:61_1"><i><font color="Green" title="E45">A14</font></i></a>, <a class="txt" href="lfuzzy_1.html#E7:61_1"><i><font color="Green" title="E46">A17</font></i></a>, <a class="txt" href="lfuzzy_1.html#E8:61_1"><i><font color="Green" title="E47">A19</font></i></a>, <a class="ref" href="lfuzzy_1.html#T32" target="_self" title="LFUZZY_1:th.32">Th32</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:61"/>
<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c3">Q</font>,<span class="p3">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p4"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span></span>)</span></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> : <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> </span>}</span> ,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="lfuzzy_1.html#T2" target="_self" title="LFUZZY_1:th.2">Th2</a></i>;<br/>


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