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

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



<b>set </b><font color="Maroon">Q</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> ;<br/>
<b>set </b><font color="Maroon">FL</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<a NAME="E1:63"/><i><font color="Green">E49</font></i>: 
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> ,<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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lfuzzy_0.html#K5">@</a> <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">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="lfuzzy_0.html#D5">LFUZZY_0:def 5</a></i>;<br/>
<a NAME="E2:63"/><i><font color="Green">E50</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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="E1:63_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:63_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>
<a NAME="E4:63_1"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> <a href="lfuzzy_0.html#K6">@</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</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="E5:63_1"/>
<font color="Maroon">q</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:63"/><i><font color="Green">E52</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p3">(<span class="default"><a href="lfuzzy_1.html#K5" target="_self">TrCl</a> <font color="Maroon">R</font></span>)</span></span>)</span> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">r'</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p5">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> ,<span class="p3">(<span class="default"><a href="lfuzzy_0.html#K4">FuzzyLattice</a> <span class="p4"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span> where <font color="Olive">r'</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">r'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p3">(<span class="default"><a href="lfuzzy_1.html#K5" target="_self">TrCl</a> <font color="Maroon">R</font></span>)</span></span>)</span> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> ;<br/>
<b>set </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">r'</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p5">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> ,<span class="p3">(<span class="default"><a href="lfuzzy_0.html#K4">FuzzyLattice</a> <span class="p4"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span> where <font color="Olive">r'</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">r'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> ;<br/>
<b>thus </b><a NAME="E1:63_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p3">(<span class="default"><a href="lfuzzy_1.html#K5" target="_self">TrCl</a> <font color="Maroon">R</font></span>)</span></span>)</span> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">r'</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p5">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> ,<span class="p3">(<span class="default"><a href="lfuzzy_0.html#K4">FuzzyLattice</a> <span class="p4"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span> where <font color="Olive">r'</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">r'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:63_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p3">(<span class="default"><a href="lfuzzy_1.html#K5" target="_self">TrCl</a> <font color="Maroon">R</font></span>)</span></span>)</span> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span><b> such that </b><br/><a NAME="E3:63_2_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Maroon">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K5" target="_self">TrCl</a> <font color="Maroon">R</font></span>)</span> &amp; <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  )
 ;<br/>
<a NAME="E4:63_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Maroon">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> ,<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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 
<b>by </b><i>, , , <a class="txt" href="lfuzzy_1.html#E1"><i><font color="Green">Lemma39</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:63_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">r'</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p5">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> ,<span class="p3">(<span class="default"><a href="lfuzzy_0.html#K4">FuzzyLattice</a> <span class="p4"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span> where <font color="Olive">r'</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">r'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> 
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E2:63_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">r'</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p5">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> ,<span class="p3">(<span class="default"><a href="lfuzzy_0.html#K4">FuzzyLattice</a> <span class="p4"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span> where <font color="Olive">r'</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">r'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p3">(<span class="default"><a href="lfuzzy_1.html#K5" target="_self">TrCl</a> <font color="Maroon">R</font></span>)</span></span>)</span> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:63_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">r'</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p5">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> ,<span class="p3">(<span class="default"><a href="lfuzzy_0.html#K4">FuzzyLattice</a> <span class="p4"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span> where <font color="Olive">r'</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">r'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span><b> such that </b><br/><a NAME="E3:63_2_2"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Maroon">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> ,<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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> &amp; <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  )
 ;<br/>
<a NAME="E4:63_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Maroon">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K5" target="_self">TrCl</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i>, , <a class="txt" href="lfuzzy_1.html#E1"><i><font color="Green">Lemma39</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E5:63_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p3">(<span class="default"><a href="lfuzzy_1.html#K5" target="_self">TrCl</a> <font color="Maroon">R</font></span>)</span></span>)</span> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> 
 <b>by </b><i/>;<br/>


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


</div><b>end;</b></div>
<a NAME="E4:63"/><i><font color="Green">E77</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>, <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">i</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p3">(<span class="default"><font color="Olive">j</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>, <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">i</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p3">(<span class="default"><font color="Olive">j</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> ;<br/>
<b>thus </b><a NAME="E1:63_3"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>, <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">i</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p3">(<span class="default"><font color="Olive">j</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:63_3_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>, <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>, <font color="Maroon">s</font> being   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span><b> such that </b><br/><a NAME="E3:63_3_1"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Maroon">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Maroon">s</font></span>)</span> &amp; <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  &amp; <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  )
 ;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:63_3_1"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i><a class="ref" href="lfuzzy_1.html#D1" target="_self">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:63_3_1"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i><a class="ref" href="lfuzzy_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E8:63_3_1"/>
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="lfuzzy_0.html#K5">@</a> <font color="Maroon">r</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="lfuzzy_0.html#K5">@</a> <font color="Maroon">s</font> )
 
<b>by </b><i><a class="ref" href="lfuzzy_0.html#D5">LFUZZY_0:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:63_3_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">i</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p3">(<span class="default"><font color="Olive">j</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="lfuzzy_1.html#D1" target="_self">Def1</a>, , </i>;<br/>


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

<b>thus </b><a NAME="E2:63_3"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">i</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p3">(<span class="default"><font color="Olive">j</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>, <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</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="E1:63_3_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">i</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p3">(<span class="default"><font color="Olive">j</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:63_3_2"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>
<a NAME="E4:63_3_2"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">i</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  &amp; <font color="Maroon">j</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  )
 
<b>by </b><i><a class="ref" href="lfuzzy_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E5:63_3_2"/><i><font color="Green">E131</font></i>: 
( <font color="Maroon">i</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> <a href="lfuzzy_0.html#K6">@</a>  &amp; <font color="Maroon">j</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> <a href="lfuzzy_0.html#K6">@</a>  )
 
<b>by </b><i><a class="ref" href="lfuzzy_0.html#D6">LFUZZY_0:def 6</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">r</font> = <font color="Maroon">i</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font> as   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">j</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font> as   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E8:63_3_2"/>
(  <a href="lfuzzy_0.html#K5">@</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp;  <a href="lfuzzy_0.html#K5">@</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> )
 
<b>by </b><i><a class="ref" href="lfuzzy_0.html#D5">LFUZZY_0:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:63_3_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>, <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  ) </span>}</span> 
 <b>by </b><i><a class="ref" href="lfuzzy_1.html#D1" target="_self">Def1</a>, </i>;<br/>


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


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>,  <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of 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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p2">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Maroon">a<sub>2</sub></font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K6">@</a> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> ;<br/>
<a NAME="E5:63"/><i><font color="Green">E132</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p5">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> ,<span class="p3">(<span class="default"><a href="lfuzzy_0.html#K4">FuzzyLattice</a> <span class="p4"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">r'</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p0">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s'</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K6">@</a> </span>)</span> where <font color="Olive">s'</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">s'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> ,<span class="p3">(<span class="default"><a href="lfuzzy_0.html#K4">FuzzyLattice</a> <span class="p4"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span> where <font color="Olive">r'</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">r'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_4"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of 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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> =  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> ,<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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of 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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> =  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p4">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K6">@</a> </span>)</span> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> ,<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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>;<br/>
<a NAME="E1:63_4"/>
for <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> holds   <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> ,<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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p4">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K6">@</a> </span>)</span> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> ,<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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>;<br/>

<a NAME="E1:63_4_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Maroon">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Maroon">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p4">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K6">@</a> </span>)</span> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_4_1_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(  <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>) <b>-&gt; </b>   <a href="fuzzy_1.html#M1">Membership_Func</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span> = <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Maroon">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(  <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of 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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Maroon">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p2">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K6">@</a> ;<br/>
<a NAME="E1:63_4_1_1"/><i><font color="Green">E133</font></i>: 
for <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> holds  <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">s</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">s</font>)
 
<b>by </b><i><a class="ref" href="lfuzzy_0.html#D6">LFUZZY_0:def 6</a></i>;<br/>
<a NAME="E2:63_4_1_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">s</font>) where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">s</font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">s</font>) where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">s</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S5">FRAENKEL:sch 5</a>();<br/></i>
<b>hence </b><a NAME="E3:63_4_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Maroon">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Maroon">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p4">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K6">@</a> </span>)</span> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:63_4_1"/>
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Maroon">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> ,<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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Maroon">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p4">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K6">@</a> </span>)</span> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> ,<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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:63_4"/><b>then </b><i><font color="Green">E134</font></i>: 
for <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> holds  <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">r</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">r</font>)
 
;<br/>
<a NAME="E3:63_4"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">r</font>) where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">r</font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">r</font>) where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">r</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S5">FRAENKEL:sch 5</a>();<br/></i>
<b>hence </b><a NAME="E4:63_4"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p5">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> ,<span class="p3">(<span class="default"><a href="lfuzzy_0.html#K4">FuzzyLattice</a> <span class="p4"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">r'</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p0">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s'</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K6">@</a> </span>)</span> where <font color="Olive">s'</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">s'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> ,<span class="p3">(<span class="default"><a href="lfuzzy_0.html#K4">FuzzyLattice</a> <span class="p4"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span> where <font color="Olive">r'</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Olive">r'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:63"/><i><font color="Green">E135</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p4">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K6">@</a> </span>)</span> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>, <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>, <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_5"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>,  <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  );<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>,  <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of 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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p2">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Maroon">a<sub>2</sub></font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K6">@</a> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>,  <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>) <b>-&gt; </b>   <a href="fuzzy_1.html#M1">Membership_Func</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span> = <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Maroon">a<sub>2</sub></font></span>)</span>;<br/>
<a NAME="E1:63_5"/><i><font color="Green">E136</font></i>: 
for <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> holds  <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">r</font>,<font color="Olive">s</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">r</font>,<font color="Olive">s</font>)
 
<b>by </b><i><a class="ref" href="lfuzzy_0.html#D6">LFUZZY_0:def 6</a></i>;<br/>
<a NAME="E2:63_5"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">r</font>,<font color="Olive">s</font>) where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>, <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">r</font>,<font color="Olive">s</font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">r</font>,<font color="Olive">s</font>) where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>, <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">r</font>,<font color="Olive">s</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S7">FRAENKEL:sch 7</a>();<br/></i>
<b>hence </b><a NAME="E3:63_5"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p4">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K6">@</a> </span>)</span> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>, <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">r</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p3">(<span class="default"><a href="lfuzzy_0.html#K5">@</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>, <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span>  ) </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><span class="p1">(<span class="default"><a href="lfuzzy_1.html#K5" target="_self">TrCl</a> <font color="Maroon">R</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K5" target="_self">TrCl</a> <font color="Maroon">R</font></span>)</span> = 
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p3">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">r</font>,<font color="Olive">s</font>) where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">s</font>] </span>}</span> ,<span class="p3">(<span class="default"><a href="lfuzzy_0.html#K4">FuzzyLattice</a> <span class="p4"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font>] </span>}</span> ,<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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
<b>by </b><i>, , , <a class="ref" href="lfuzzy_1.html#D1" target="_self">Def1</a>, </i>
<br/>.= 
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">r</font>,<font color="Olive">s</font>) where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>, <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">s</font>] ) </span>}</span> ,<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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
<b>from </b><i><a class="ref" href="lfuzzy_0.html#S1">LFUZZY_0:sch 1</a>()</i>
<br/>.= 
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">i</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p3">(<span class="default"><font color="Olive">j</font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">R</font></span>)</span></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> ,<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">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
<b>by </b><i>, </i>
;<br/>


</div>
