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

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">Y</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">Z</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">R</font> be   <a href="fuzzy_3.html#NM1">RMembership_Func</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">S</font> be   <a href="fuzzy_3.html#NM1">RMembership_Func</a> of <font color="Maroon">Y</font>,<font color="Maroon">Z</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Z</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" target="_self">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span>;<br/>











<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Maroon">x</font>,<font color="Olive">y</font></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Olive">y</font>,<font color="Maroon">z</font></span>)</span></span>)</span> where <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon">B</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"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Maroon">x</font>,<font color="Olive">y</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p4">(<span class="default"><font color="Maroon">S</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Olive">y</font>,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">a</font></span>)</span> where <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : verum </span>}</span> ;<br/>
<a NAME="E1:63"/>
for <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Maroon">x</font>,<font color="Olive">y</font></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Olive">y</font>,<font color="Maroon">z</font></span>)</span></span>)</span> where <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : verum </span>}</span>  iff <font color="Olive">b</font> <a href="hidden.html#R2">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"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Maroon">x</font>,<font color="Olive">y</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p4">(<span class="default"><font color="Maroon">S</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Olive">y</font>,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">a</font></span>)</span> where <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : verum </span>}</span>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:63_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Maroon">x</font>,<font color="Olive">y</font></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Olive">y</font>,<font color="Maroon">z</font></span>)</span></span>)</span> where <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : verum </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:63_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>
 ;<br/><a NAME="E4:63_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#T16">LATTICE3:16</a></i>;<br/><b>hence </b><a NAME="E5:63_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">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"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Maroon">x</font>,<font color="Olive">y</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p4">(<span class="default"><font color="Maroon">S</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Olive">y</font>,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">a</font></span>)</span> where <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : verum </span>}</span> 
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:63_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">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"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Maroon">x</font>,<font color="Olive">y</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p4">(<span class="default"><font color="Maroon">S</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Olive">y</font>,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">a</font></span>)</span> where <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E4:63_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">a</font>
 ;<br/>
<a NAME="E5:63_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#T16">LATTICE3:16</a></i>;<br/>
<b>hence </b><a NAME="E6:63_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Maroon">x</font>,<font color="Olive">y</font></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Olive">y</font>,<font color="Maroon">z</font></span>)</span></span>)</span> where <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : verum </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:63"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Maroon">x</font>,<font color="Olive">y</font></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Olive">y</font>,<font color="Maroon">z</font></span>)</span></span>)</span> where <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : verum </span>}</span>  <a href="hidden.html#R1">=</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"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Maroon">x</font>,<font color="Olive">y</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p4">(<span class="default"><font color="Maroon">S</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Olive">y</font>,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">a</font></span>)</span> where <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<b>hence </b><a NAME="E3:63"/>
<font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">S</font></span>)</span> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <span class="p4">(<span class="default"><font color="Maroon">R</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Maroon">x</font>,<font color="Olive">y</font></span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="lfuzzy_0.html#K8" target="_self">.</a> <font color="Olive">y</font>,<font color="Maroon">z</font></span>)</span></span>)</span> where <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : verum </span>}</span> ,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" target="_self">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span>
 <b>by </b><i/>;<br/>


</div>
