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

<b>set </b><font color="Maroon" title="c3">FL</font> =  <a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c4">RP</font> =  <a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c5">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">j</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> ;<br/>
<a NAME="E1:63_1_1"/>
for <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K5" title="LFUZZY_1:func.5">TrCl</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p2">(<span class="default"><a href="lfuzzy_1.html#K5" title="LFUZZY_1:func.5">TrCl</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Olive" title="b1">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K5" title="LFUZZY_1:func.5">TrCl</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Olive" title="b1">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>

<a NAME="E1:63_1_1_1"/><i><font color="Green" title="E41">A1</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">j</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">t</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:63_1_1_1_1"/>
<font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">j</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">i</font>, <font color="Maroon" title="c9">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:63_1_1_1_1"/><i><font color="Green" title="E42">A2</font></i>: 
( <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span> &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>
<a NAME="E4:63_1_1_1_1"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K6" title="LFUZZY_0:func.6">@</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="lfuzzy_0.html#D6" title="LFUZZY_0:def.6">LFUZZY_0:def 6</a></i>;<br/>
<b>hence </b><a NAME="E5:63_1_1_1_1"/>
<font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="lfuzzy_1.html#E3:63_1_1_1_1"><i><font color="Green" title="E42">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green" title="E42">A3</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K5" title="LFUZZY_1:func.5">TrCl</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p2">(<span class="default"><a href="lfuzzy_1.html#K5" title="LFUZZY_1:func.5">TrCl</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Maroon" title="c6">c</font> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">i</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p4">(<span class="default"><font color="Olive" title="b2">j</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> ,<span class="p2">(<span class="default"><a href="lfuzzy_0.html#K4" title="LFUZZY_0:func.4">FuzzyLattice</a> <span class="p3"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span></span>)</span> <a href="lfuzzy_0.html#K9" title="LFUZZY_0:func.9">.</a> <font color="Maroon" title="c6">c</font>
<b>by </b><i><a class="ref" href="lfuzzy_1.html#T36" target="_self" title="LFUZZY_1:th.36">Th36</a></i>
<br/>.= 
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">(<span class="default"><a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">i</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p4">(<span class="default"><font color="Olive" title="b2">j</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> ,<font color="Maroon" title="c6">c</font></span>)</span>,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E1:63_1_1_1"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="lfuzzy_1.html#T32" target="_self" title="LFUZZY_1:th.32">Th32</a></i>
;<br/>
<a NAME="E3:63_1_1_1"/>
<span class="p1">(<span class="default"><a href="lfuzzy_1.html#K5" title="LFUZZY_1:func.5">TrCl</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Maroon" title="c6">c</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">j</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> ,<font color="Maroon" title="c6">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:63_1_1_1_3"/>
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b2">i</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">j</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span> where <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> ,<font color="Maroon" title="c6">c</font> holds <br/><font color="Olive" title="b1">b</font> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&lt;=</a> <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K5" title="LFUZZY_1:func.5">TrCl</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Maroon" title="c6">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:63_1_1_1_3_1"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">j</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> ,<font color="Maroon" title="c6">c</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:63_1_1_1_3_1"/><i><font color="Green" title="E43">A4</font></i>: 
( <font color="Maroon" title="c8">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">j</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span>  &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">c</font> )
 <b>by </b><i><a class="ref" href="card_3.html#D6" title="CARD_3:def.6">CARD_3:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c10">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:63_1_1_1_3_1"/><i><font color="Green" title="E44">A5</font></i>: 
( <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span> &amp; <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="lfuzzy_1.html#E3:63_1_1_1_3_1"><i><font color="Green" title="E43">A4</font></i></a></i>;<br/>
<a NAME="E6:63_1_1_1_3_1"/><i><font color="Green" title="E45">A6</font></i>: 
<font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E5:63_1_1_1_3_1"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="xreal_1.html#T36" title="XREAL_1:th.36">XREAL_1:36</a></i>;<br/>
<a NAME="E7:63_1_1_1_3_1"/><i><font color="Green" title="E46">A7</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">j</font></span>)</span> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font>
 
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E5:63_1_1_1_3_1"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="lfuzzy_1.html#T27" target="_self" title="LFUZZY_1:th.27">Th27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">f</font> = <font color="Maroon" title="c8">f</font> as   <a href="fuzzy_3.html#NM1" title="FUZZY_3:NM.1">RMembership_Func</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="lfuzzy_1.html#E5:63_1_1_1_3_1"><i><font color="Green" title="E44">A5</font></i></a></i>;<br/>
<a NAME="E9:63_1_1_1_3_1"/>
 <a href="lfuzzy_1.html#K5" title="LFUZZY_1:func.5">TrCl</a> <font color="Maroon" title="c2">R</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a> 
 
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E6:63_1_1_1_3_1"><i><font color="Green" title="E45">A6</font></i></a>, <a class="txt" href="lfuzzy_1.html#E7:63_1_1_1_3_1"><i><font color="Green" title="E46">A7</font></i></a>, <a class="ref" href="lfuzzy_1.html#T31" target="_self" title="LFUZZY_1:th.31">Th31</a></i>;<br/>
<a NAME="E10:63_1_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c11">f</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Maroon" title="c6">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K5" title="LFUZZY_1:func.5">TrCl</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Maroon" title="c6">c</font>
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D3" title="FUZZY_1:def.3">FUZZY_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E11:63_1_1_1_3_1"/>
<font color="Maroon" title="c7">b</font> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&lt;=</a> <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K5" title="LFUZZY_1:func.5">TrCl</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Maroon" title="c6">c</font>
 <b>by </b><i><a class="txt" href="lfuzzy_1.html#E3:63_1_1_1_3_1"><i><font color="Green" title="E43">A4</font></i></a>, <a class="ref" href="lfuzzy_0.html#T3" title="LFUZZY_0:th.3">LFUZZY_0:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:63_1_1_1_3"/>
<span class="p1">(<span class="default"><a href="lfuzzy_1.html#K5" title="LFUZZY_1:func.5">TrCl</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Maroon" title="c6">c</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">j</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> ,<font color="Maroon" title="c6">c</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:63_1_1_1"/><b>then </b>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">(<span class="default"><a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">i</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p4">(<span class="default"><font color="Olive" title="b2">j</font> <a href="lfuzzy_1.html#K4" title="LFUZZY_1:func.4">iter</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> ,<font color="Maroon" title="c6">c</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="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&lt;=</a> <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K5" title="LFUZZY_1:func.5">TrCl</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Maroon" title="c6">c</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T32" title="YELLOW_0:th.32">YELLOW_0:32</a></i>;<br/>
<b>hence </b><a NAME="E5:63_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K5" title="LFUZZY_1:func.5">TrCl</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p2">(<span class="default"><a href="lfuzzy_1.html#K5" title="LFUZZY_1:func.5">TrCl</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Maroon" title="c6">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K5" title="LFUZZY_1:func.5">TrCl</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Maroon" title="c6">c</font>
 <b>by </b><i><a class="txt" href="lfuzzy_1.html#E2:63_1_1_1"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="lfuzzy_0.html#T3" title="LFUZZY_0:th.3">LFUZZY_0:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:63_1_1"/><b>then </b>
 <a href="lfuzzy_1.html#K5" title="LFUZZY_1:func.5">TrCl</a> <font color="Maroon" title="c2">R</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a> 
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D3" title="FUZZY_1:def.3">FUZZY_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:63_1_1"/>
 <a href="lfuzzy_1.html#K5" title="LFUZZY_1:func.5">TrCl</a> <font color="Maroon" title="c2">R</font> is <a href="lfuzzy_1.html#V3" title="LFUZZY_1:attr.3">transitive</a>
 <b>by </b><i><a class="ref" href="lfuzzy_1.html#D6" target="_self" title="LFUZZY_1:def.6">Def6</a></i>;<br/>


</div>
