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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E1:32"/>
 <a href="fuzzy_2.html#K5" title="FUZZY_2:func.5">Umf</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> is <a href="lfuzzy_1.html#V2" title="LFUZZY_1:attr.2">symmetric</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lfuzzy_1.html#D5" target="_self" title="LFUZZY_1:def.5">LFUZZY_1:def 5</a><br/>

<b>thus </b><span class="p1">(<span class="default"><a href="fuzzy_2.html#K5" title="FUZZY_2:func.5">Umf</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> = 
<span class="p1">(<span class="default"><a href="fuzzy_2.html#K5" title="FUZZY_2:func.5">Umf</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>

<br/>.= 
1
<b>by </b><i><a class="ref" href="fuzzy_4.html#T21" title="FUZZY_4:th.21">FUZZY_4:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fuzzy_2.html#K5" title="FUZZY_2:func.5">Umf</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="ref" href="fuzzy_4.html#T21" title="FUZZY_4:th.21">FUZZY_4:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fuzzy_2.html#K5" title="FUZZY_2:func.5">Umf</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font>

;<br/>


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

<b>thus </b><a NAME="E2:32"/>
 <a href="fuzzy_2.html#K5" title="FUZZY_2:func.5">Umf</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> is <a href="lfuzzy_1.html#V3" title="LFUZZY_1:attr.3">transitive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lfuzzy_1.html#D7" target="_self" title="LFUZZY_1:def.7">LFUZZY_1:def 7</a><br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_2.html#K5" title="FUZZY_2:func.5">Umf</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_2.html#K5" title="FUZZY_2:func.5">Umf</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> = 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> 1,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_2.html#K5" title="FUZZY_2:func.5">Umf</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="fuzzy_4.html#T20" title="FUZZY_4:th.20">FUZZY_4:20</a></i>
<br/>.= 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> 1,1
<b>by </b><i><a class="ref" href="fuzzy_4.html#T20" title="FUZZY_4:th.20">FUZZY_4:20</a></i>
<br/>.= 
1

;<br/>
<a NAME="E2:32_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_2.html#K5" title="FUZZY_2:func.5">Umf</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_2.html#K5" title="FUZZY_2:func.5">Umf</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_2.html#K5" title="FUZZY_2:func.5">Umf</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="ref" href="fuzzy_4.html#T20" title="FUZZY_4:th.20">FUZZY_4:20</a></i>;<br/>
<b>hence </b><a NAME="E3:32_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_2.html#K5" title="FUZZY_2:func.5">Umf</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_2.html#K5" title="FUZZY_2:func.5">Umf</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_2.html#K5" title="FUZZY_2:func.5">Umf</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="ref" href="lfuzzy_0.html#T3" title="LFUZZY_0:th.3">LFUZZY_0:3</a></i>;<br/>


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

<b>thus </b><a NAME="E3:32"/>
 <a href="fuzzy_2.html#K5" title="FUZZY_2:func.5">Umf</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> is <a href="lfuzzy_1.html#V1" title="LFUZZY_1:attr.1">reflexive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lfuzzy_1.html#D3" target="_self" title="LFUZZY_1:def.3">LFUZZY_1:def 3</a><br/>

<a NAME="E1:32_3"/>
<span class="p1">(<span class="default"><a href="fuzzy_2.html#K5" title="FUZZY_2:func.5">Umf</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="fuzzy_4.html#T21" title="FUZZY_4:th.21">FUZZY_4:21</a></i>;<br/>
<b>hence </b><a NAME="E2:32_3"/>
<span class="p1">(<span class="default"><a href="fuzzy_2.html#K5" title="FUZZY_2:func.5">Umf</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>


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


</div>
