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

<b>thus </b><a NAME="E1:20_1_1"/>
 <a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font> is <a href="lfuzzy_1.html#V2" target="_self">symmetric</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lfuzzy_1.html#D5" target="_self">LFUZZY_1:def 5</a><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or  not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:20_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:20_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="domain_1.html#K1">]</a></span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:20_1_1_1_1_2"/><i><font color="Green">E49</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> = 
0
<b>by </b><i><a class="ref" href="fuzzy_4.html#D4">FUZZY_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, <a class="ref" href="fuzzy_4.html#D4">FUZZY_4:def 4</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E2:20_1_1"/>
 <a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font> is <a href="lfuzzy_1.html#V3" target="_self">transitive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lfuzzy_1.html#D7" target="_self">LFUZZY_1:def 7</a><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> or  not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:20_1_1_2_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:20_1_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1_1_2_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or  not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:20_1_1_2_1_1_1_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> = 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="lfuzzy_0.html#T5">LFUZZY_0:5</a></i>
<br/>.= 
1
<b>by </b><i>, , <a class="ref" href="fuzzy_4.html#D4">FUZZY_4:def 4</a></i>
;<br/><a NAME="E3:20_1_1_2_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i>, <a class="ref" href="fuzzy_4.html#D4">FUZZY_4:def 4</a></i>;<br/><b>hence </b><a NAME="E4:20_1_1_2_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="lfuzzy_0.html#T3">LFUZZY_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:20_1_1_2_1_1_1_1_2"/><i><font color="Green">E52</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> = 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="lfuzzy_0.html#T5">LFUZZY_0:5</a></i>
<br/>.= 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,0
<b>by </b><i>, , <a class="ref" href="fuzzy_4.html#D4">FUZZY_4:def 4</a></i>
<br/>.= 
 <a href="xxreal_0.html#K3">min</a> 0,0
<b>by </b><i>, <a class="ref" href="fuzzy_4.html#D4">FUZZY_4:def 4</a></i>
<br/>.= 
0

;<br/><a NAME="E3:20_1_1_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/><a NAME="E4:20_1_1_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i>, <a class="ref" href="fuzzy_4.html#D4">FUZZY_4:def 4</a></i>;<br/><b>hence </b><a NAME="E5:20_1_1_2_1_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="lfuzzy_0.html#T3">LFUZZY_0:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:20_1_1_2_1_2"/><i><font color="Green">E53</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:20_1_1_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1_1_2_1_2_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:20_1_1_2_1_2_1_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> = 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="lfuzzy_0.html#T5">LFUZZY_0:5</a></i>
<br/>.= 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,0
<b>by </b><i>, , <a class="ref" href="fuzzy_4.html#D4">FUZZY_4:def 4</a></i>
<br/>.= 
 <a href="xxreal_0.html#K3">min</a> 1,0
<b>by </b><i>, <a class="ref" href="fuzzy_4.html#D4">FUZZY_4:def 4</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>
;<br/><a NAME="E3:20_1_1_2_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i>, <a class="ref" href="fuzzy_4.html#D4">FUZZY_4:def 4</a></i>;<br/><b>hence </b><a NAME="E4:20_1_1_2_1_2_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="lfuzzy_0.html#T3">LFUZZY_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:20_1_1_2_1_2_1_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:20_1_1_2_1_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_1_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1_1_2_1_2_1_1_2_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> or <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:20_1_1_2_1_2_1_1_2_1_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> = 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="lfuzzy_0.html#T5">LFUZZY_0:5</a></i>
<br/>.= 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,1
<b>by </b><i>, <a class="ref" href="fuzzy_4.html#D4">FUZZY_4:def 4</a></i>
<br/>.= 
 <a href="xxreal_0.html#K3">min</a> 0,1
<b>by </b><i>, <a class="ref" href="fuzzy_4.html#D4">FUZZY_4:def 4</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>
;<br/><a NAME="E3:20_1_1_2_1_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i>, <a class="ref" href="fuzzy_4.html#D4">FUZZY_4:def 4</a></i>;<br/><b>hence </b><a NAME="E4:20_1_1_2_1_2_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="lfuzzy_0.html#T3">LFUZZY_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:20_1_1_2_1_2_1_1_2_1_1_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> = 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="lfuzzy_0.html#T5">LFUZZY_0:5</a></i>
<br/>.= 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,0
<b>by </b><i>, <a class="ref" href="fuzzy_4.html#D4">FUZZY_4:def 4</a></i>
<br/>.= 
 <a href="xxreal_0.html#K3">min</a> 0,0
<b>by </b><i>, <a class="ref" href="fuzzy_4.html#D4">FUZZY_4:def 4</a></i>
<br/>.= 
0

;<br/><a NAME="E3:20_1_1_2_1_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i>, <a class="ref" href="fuzzy_4.html#D4">FUZZY_4:def 4</a></i>;<br/><b>hence </b><a NAME="E4:20_1_1_2_1_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="lfuzzy_0.html#T3">LFUZZY_0:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

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

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

<b>thus </b><a NAME="E3:20_1_1"/>
 <a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font> is <a href="lfuzzy_1.html#V1" target="_self">reflexive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:20_1_1_3"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds  <span class="p1">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">x</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="fuzzy_4.html#D4">FUZZY_4:def 4</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="lfuzzy_1.html#D3" target="_self">LFUZZY_1:def 3</a><br/>


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

<b>thus </b><a NAME="E4:20_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <span class="p1">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/><span class="p1">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">x</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="fuzzy_4.html#D4">FUZZY_4:def 4</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="lfuzzy_1.html#D9" target="_self">LFUZZY_1:def 9</a><br/>


</div>
