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

<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font>, <font color="Maroon" title="c3">Z</font>, <font color="Maroon" title="c4">W</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">R</font> being   <a href="fuzzy_2.html#NM1" title="FUZZY_2:NM.1">RMembership_Func</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font><br/>  for <font color="Olive" title="b2">S</font> being   <a href="fuzzy_2.html#NM1" title="FUZZY_2:NM.1">RMembership_Func</a> of <font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c3">Z</font><br/>  for <font color="Olive" title="b3">T</font> being   <a href="fuzzy_2.html#NM1" title="FUZZY_2:NM.1">RMembership_Func</a> of <font color="Maroon" title="c3">Z</font>,<font color="Maroon" title="c4">W</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Olive" title="b2">S</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Olive" title="b3">T</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Olive" title="b3">T</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">R</font> be   <a href="fuzzy_2.html#NM1" title="FUZZY_2:NM.1">RMembership_Func</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="fuzzy_2.html#NM1" title="FUZZY_2:NM.1">RMembership_Func</a> of <font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c3">Z</font><br/>  for <font color="Olive" title="b2">T</font> being   <a href="fuzzy_2.html#NM1" title="FUZZY_2:NM.1">RMembership_Func</a> of <font color="Maroon" title="c3">Z</font>,<font color="Maroon" title="c4">W</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Olive" title="b2">T</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Olive" title="b2">T</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">S</font> be   <a href="fuzzy_2.html#NM1" title="FUZZY_2:NM.1">RMembership_Func</a> of <font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c3">Z</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">T</font> being   <a href="fuzzy_2.html#NM1" title="FUZZY_2:NM.1">RMembership_Func</a> of <font color="Maroon" title="c3">Z</font>,<font color="Maroon" title="c4">W</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Olive" title="b1">T</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Olive" title="b1">T</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">T</font> be   <a href="fuzzy_2.html#NM1" title="FUZZY_2:NM.1">RMembership_Func</a> of <font color="Maroon" title="c3">Z</font>,<font color="Maroon" title="c4">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c7">T</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c7">T</font></span>)</span></span><br/>







<a NAME="E1:62"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">W</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c7">T</font></span>)</span> <a href="fuzzy_4.html#K1" title="FUZZY_4:func.1">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c7">T</font></span>)</span></span>)</span> <a href="fuzzy_4.html#K1" title="FUZZY_4:func.1">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">w</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c9">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">W</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c7">T</font></span>)</span> <a href="fuzzy_4.html#K1" title="FUZZY_4:func.1">.</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c7">T</font></span>)</span></span>)</span> <a href="fuzzy_4.html#K1" title="FUZZY_4:func.1">.</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">w</font>) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:62_1"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">and </span><br/><a NAME="E2:62_1"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
<font color="Maroon" title="c9">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c7">T</font></span>)</span> <a href="fuzzy_4.html#K1" title="FUZZY_4:func.1">.</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c7">T</font></span>)</span></span>)</span> <a href="fuzzy_4.html#K1" title="FUZZY_4:func.1">.</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">w</font>)</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c10">w</font> = <font color="Maroon" title="c9">w</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">W</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_1"><span class="lab"><font color="Green" title="E30">A3</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">x</font> = <font color="Maroon" title="c8">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">A</font> = <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"><font color="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b2">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">z</font>)</span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">z</font>,<font color="Maroon" title="c10">w</font>)</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="c2">Y</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">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</font> : verum </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">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" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">z</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">z</font>,<font color="Maroon" title="c10">w</font>)</span>)</span></span>)</span> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</font> : verum </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">C</font> = <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"><font color="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">z</font>)</span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b2">z</font>,<font color="Maroon" title="c10">w</font>)</span>)</span></span>)</span> where <font color="Olive" title="b2">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</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">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : verum </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">D</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c7">T</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c10">w</font>)</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="c2">Y</font> : verum </span> } </span> ;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>verum;<br/>
<span class="kw">deffunc </span><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="c2">Y</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</font>) <span class="kw">-&gt; </span>   <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"><font color="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,$1)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> ($1,$2)</span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> ($2,<font color="Maroon" title="c10">w</font>)</span>)</span>;<br/>
<a NAME="E5:62_1"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
 for <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">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"><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"><font color="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b3">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b3">y</font>,<font color="Olive" title="b2">z</font>)</span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b2">z</font>,<font color="Maroon" title="c10">w</font>)</span>)</span></span>)</span> where <font color="Olive" title="b3">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</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="b2">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</font> : verum </span> } </span>  iff <font color="Olive" title="b1">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"><font color="Maroon" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b2">z</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b2">z</font>,<font color="Maroon" title="c10">w</font>)</span>)</span></span>)</span> where <font color="Olive" title="b2">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</font> : verum </span> } </span>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">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"><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"><font color="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b2">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">z</font>)</span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">z</font>,<font color="Maroon" title="c10">w</font>)</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="c2">Y</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">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</font> : verum </span> } </span>  iff <font color="Maroon" title="c16">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"><font color="Maroon" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">z</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">z</font>,<font color="Maroon" title="c10">w</font>)</span>)</span></span>)</span> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</font> : verum </span> } </span>  )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">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"><font color="Maroon" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">z</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">z</font>,<font color="Maroon" title="c10">w</font>)</span>)</span></span>)</span> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</font> : verum </span> } </span>  implies <font color="Maroon" title="c16">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"><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"><font color="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b2">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">z</font>)</span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">z</font>,<font color="Maroon" title="c10">w</font>)</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="c2">Y</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">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</font> : verum </span> } </span>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:62_1_1_1"/>
<font color="Maroon" title="c16">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"><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"><font color="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b2">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">z</font>)</span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">z</font>,<font color="Maroon" title="c10">w</font>)</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="c2">Y</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">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</font> : verum </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">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"><font color="Maroon" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">z</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">z</font>,<font color="Maroon" title="c10">w</font>)</span>)</span></span>)</span> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</font> : verum </span> } </span> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</font><span class="kw"> such that </span><br/><a NAME="E3:62_1_1_1"/><span class="lab"><font color="Green" title="E32">A5</font></span>: 
<font color="Maroon" title="c16">a</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"><span class="p4">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c17">z</font>)</span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c17">z</font>,<font color="Maroon" title="c10">w</font>)</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="c2">Y</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="E4:62_1_1_1"/>
<font color="Maroon" title="c16">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c17">z</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c17">z</font>,<font color="Maroon" title="c10">w</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_1_1_1"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="txt" href="lfuzzy_0.html#E34"><span class="lab"><font color="Green" title="E26">Lm6</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:62_1_1_1"/>
<font color="Maroon" title="c16">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"><font color="Maroon" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">z</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">z</font>,<font color="Maroon" title="c10">w</font>)</span>)</span></span>)</span> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</font> : verum </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:62_1_1"/>
<font color="Maroon" title="c16">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"><font color="Maroon" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">z</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">z</font>,<font color="Maroon" title="c10">w</font>)</span>)</span></span>)</span> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</font> : verum </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">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"><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"><font color="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b2">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">z</font>)</span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">z</font>,<font color="Maroon" title="c10">w</font>)</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="c2">Y</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">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</font> : verum </span> } </span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</font><span class="kw"> such that </span><br/><a NAME="E4:62_1_1"/><span class="lab"><font color="Green" title="E32">A6</font></span>: 
<font color="Maroon" title="c16">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c17">z</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c17">z</font>,<font color="Maroon" title="c10">w</font>)</span>)</span>
 ;<br/>
<a NAME="E5:62_1_1"/>
<font color="Maroon" title="c16">a</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"><span class="p4">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c17">z</font>)</span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c17">z</font>,<font color="Maroon" title="c10">w</font>)</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="c2">Y</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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62_1_1"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="lfuzzy_0.html#E34"><span class="lab"><font color="Green" title="E26">Lm6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:62_1_1"/>
<font color="Maroon" title="c16">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"><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"><font color="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b2">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">z</font>)</span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">z</font>,<font color="Maroon" title="c10">w</font>)</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="c2">Y</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">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</font> : verum </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:62_1"/><span class="lab"><font color="Green" title="E32">A7</font></span>: 
 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="c2">Y</font><br/>  for <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="c3">Z</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">z</font>] holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">z</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">z</font>)
 
;<br/>
<a NAME="E7:62_1"/><span class="lab"><font color="Green" title="E33">A8</font></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"><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">z</font>) where <font color="Olive" title="b2">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">z</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="c2">Y</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>1</sub></font>(<font color="Olive" title="b2">y1</font>,<font color="Olive" title="b1">z1</font>) where <font color="Olive" title="b2">y1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">y1</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">z1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">z1</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>)
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="lfuzzy_0.html#S5" target="_self" title="LFUZZY_0:sch.5">LFUZZY_0:sch 5</a>(<a class="txt" href="#E6:62_1"><span class="lab"><font color="Green" title="E32">A7</font></span></a>);<br/></span>
<a NAME="E8:62_1"/>
 for <font color="Olive" title="b1">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><font color="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b2">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>)</span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b3">z</font>,<font color="Maroon" title="c10">w</font>)</span>)</span></span>)</span> where <font color="Olive" title="b3">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</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="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : verum </span> } </span>  iff <font color="Olive" title="b1">c</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="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b2">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c7">T</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b2">y</font>,<font color="Maroon" title="c10">w</font>)</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="c2">Y</font> : verum </span> } </span>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><font color="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">z</font>)</span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b2">z</font>,<font color="Maroon" title="c10">w</font>)</span>)</span></span>)</span> where <font color="Olive" title="b2">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</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">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : verum </span> } </span>  iff <font color="Maroon" title="c16">c</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="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c7">T</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c10">w</font>)</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="c2">Y</font> : verum </span> } </span>  )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">c</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="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c7">T</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c10">w</font>)</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="c2">Y</font> : verum </span> } </span>  implies <font color="Maroon" title="c16">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><font color="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">z</font>)</span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b2">z</font>,<font color="Maroon" title="c10">w</font>)</span>)</span></span>)</span> where <font color="Olive" title="b2">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</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">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : verum </span> } </span>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:62_1_2_1"/>
<font color="Maroon" title="c16">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><font color="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">z</font>)</span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b2">z</font>,<font color="Maroon" title="c10">w</font>)</span>)</span></span>)</span> where <font color="Olive" title="b2">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</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">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : verum </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">c</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="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c7">T</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c10">w</font>)</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="c2">Y</font> : verum </span> } </span> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font><span class="kw"> such that </span><br/><a NAME="E3:62_1_2_1"/><span class="lab"><font color="Green" title="E34">A9</font></span>: 
<font color="Maroon" title="c16">c</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"><span class="p4">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c17">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c17">y</font>,<font color="Olive" title="b1">z</font>)</span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">z</font>,<font color="Maroon" title="c10">w</font>)</span>)</span></span>)</span> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</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="E4:62_1_2_1"/>
<font color="Maroon" title="c16">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c17">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c7">T</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c17">y</font>,<font color="Maroon" title="c10">w</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_1_2_1"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="lfuzzy_0.html#E35"><span class="lab"><font color="Green" title="E27">Lm7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:62_1_2_1"/>
<font color="Maroon" title="c16">c</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="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c7">T</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c10">w</font>)</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="c2">Y</font> : verum </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:62_1_2"/>
<font color="Maroon" title="c16">c</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="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c7">T</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c10">w</font>)</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="c2">Y</font> : verum </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><font color="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">z</font>)</span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b2">z</font>,<font color="Maroon" title="c10">w</font>)</span>)</span></span>)</span> where <font color="Olive" title="b2">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</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">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : verum </span> } </span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font><span class="kw"> such that </span><br/><a NAME="E4:62_1_2"/><span class="lab"><font color="Green" title="E34">A10</font></span>: 
<font color="Maroon" title="c16">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c17">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c7">T</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c17">y</font>,<font color="Maroon" title="c10">w</font>)</span>)</span>
 ;<br/>
<a NAME="E5:62_1_2"/>
<font color="Maroon" title="c16">c</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"><span class="p4">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c17">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c17">y</font>,<font color="Olive" title="b1">z</font>)</span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">z</font>,<font color="Maroon" title="c10">w</font>)</span>)</span></span>)</span> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62_1_2"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="lfuzzy_0.html#E35"><span class="lab"><font color="Green" title="E27">Lm7</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:62_1_2"/>
<font color="Maroon" title="c16">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><font color="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">z</font>)</span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b2">z</font>,<font color="Maroon" title="c10">w</font>)</span>)</span></span>)</span> where <font color="Olive" title="b2">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</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">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : verum </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:62_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A11</font></span>: 
<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"><font color="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">z</font>)</span>)</span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b2">z</font>,<font color="Maroon" title="c10">w</font>)</span>)</span></span>)</span> where <font color="Olive" title="b2">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</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">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</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"><font color="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c7">T</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c10">w</font>)</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="c2">Y</font> : verum </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>;<br/>
<a NAME="E10:62_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c7">T</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c10">w</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"><span class="p4">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">z</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">z</font>,<font color="Maroon" title="c10">w</font>)</span>)</span></span>)</span> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</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>) &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c7">T</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c10">w</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"><font color="Maroon" title="c5">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">y</font>)</span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c7">T</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> (<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c10">w</font>)</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="c2">Y</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>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lfuzzy_0.html#T22" target="_self" title="LFUZZY_0:th.22">Th22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:62_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c7">T</font></span>)</span> <a href="fuzzy_4.html#K1" title="FUZZY_4:func.1">.</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">w</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c7">T</font></span>)</span></span>)</span> <a href="fuzzy_4.html#K1" title="FUZZY_4:func.1">.</a> (<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">w</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:62_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="#E9:62_1"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="txt" href="#E7:62_1"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:62"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c7">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">W</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c7">T</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">W</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:62"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">S</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c7">T</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c7">T</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="funct_3.html#T6" title="FUNCT_3:th.6">FUNCT_3:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
