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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="fuzzy_3.html#NM1">RMembership_Func</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:66"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="lfuzzy_1.html#V3" target="_self">transitive</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_1.html#NR2" target="_self">c=</a>  )
 ;<br/>

<a NAME="E2:66"/>
 <a href="lfuzzy_1.html#K5" target="_self">TrCl</a> <font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_1.html#NR2" target="_self">c=</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<a NAME="E1:66_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> holds <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K5" target="_self">TrCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K5" target="_self">TrCl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="lfuzzy_0.html#K4">FuzzyLattice</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="lfuzzy_0.html#K1">RealPoset</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>;<br/>
<a NAME="E1:66_1_1"/><i><font color="Green">E44</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:66_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:66_1_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>
<a NAME="E4:66_1_1_1"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lfuzzy_0.html#K6">@</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="lfuzzy_0.html#D6">LFUZZY_0:def 6</a></i>;<br/>
<b>hence </b><a NAME="E5:66_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K4">FuzzyLattice</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="lfuzzy_1.html#E3:66_1_1_1"><i><font color="Green">E45</font></i></a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E45</font></i>: <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K5" target="_self">TrCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> ,<span class="p2">(<span class="default"><a href="lfuzzy_0.html#K4">FuzzyLattice</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span> <a href="lfuzzy_0.html#K8">.</a> <font color="Maroon">c<sub>4</sub></font>

<br/>.= 
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><a href="card_3.html#K5">pi</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> ,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E1:66_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="lfuzzy_1.html#T32" target="_self">Th32</a></i>
;<br/>
<a NAME="E3:66_1_1"/>
<span class="p1">(<span class="default"><a href="lfuzzy_1.html#K5" target="_self">TrCl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a>  <a href="card_3.html#K5">pi</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> ,<font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:66_1_1_3"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5">pi</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> ,<font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K5" target="_self">TrCl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:66_1_1_3_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5">pi</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> ,<font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:66_1_1_3_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:66_1_1_3_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="lfuzzy_1.html#E3:66_1_1_3_1"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E6:66_1_1_3_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_1.html#NR2" target="_self">c=</a>  &amp;  <a href="lfuzzy_1.html#K5" target="_self">TrCl</a> <font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_1.html#NR2" target="_self">c=</a>  )
 
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E1:66"><i><font color="Green">E43</font></i></a>, <a class="txt" href="lfuzzy_1.html#E5:66_1_1_3_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="lfuzzy_1.html#T31" target="_self">Th31</a>, <a class="ref" href="lfuzzy_1.html#T39" target="_self">Th39</a></i>;<br/>
<a NAME="E7:66_1_1_3_1"/><b>then </b>
 <a href="lfuzzy_1.html#K5" target="_self">TrCl</a> <font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_1.html#NR2" target="_self">c=</a> 
 
<b>by </b><i><a class="ref" href="lfuzzy_1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E8:66_1_1_3_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K5" target="_self">TrCl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D3">FUZZY_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E9:66_1_1_3_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K5" target="_self">TrCl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="lfuzzy_1.html#E3:66_1_1_3_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="lfuzzy_1.html#E5:66_1_1_3_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="lfuzzy_0.html#T3">LFUZZY_0:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:66_1_1_3"/>
<span class="p1">(<span class="default"><a href="lfuzzy_1.html#K5" target="_self">TrCl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a>  <a href="card_3.html#K5">pi</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> ,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:66_1_1"/><b>then </b>
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><a href="card_3.html#K5">pi</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> ,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K5" target="_self">TrCl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>
<b>hence </b><a NAME="E5:66_1_1"/>
<span class="p1">(<span class="default"><a href="lfuzzy_1.html#K5" target="_self">TrCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K5" target="_self">TrCl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="lfuzzy_1.html#E2:66_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="lfuzzy_0.html#T3">LFUZZY_0:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:66_1"/>
 <a href="lfuzzy_1.html#K5" target="_self">TrCl</a> <font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_1.html#NR2" target="_self">c=</a> 
 <b>by </b><i><a class="ref" href="fuzzy_1.html#D3">FUZZY_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:66"/>
<font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_1.html#NR2" target="_self">c=</a> 
 <b>by </b><i><a class="txt" href="lfuzzy_1.html#E1:66"><i><font color="Green">E43</font></i></a>, <a class="ref" href="lfuzzy_1.html#T38" target="_self">Th38</a></i>;<br/>


</div>
