<?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>thus </b><a NAME="E1:32"/>
 <a href="fuzzy_3.html#K1">Zmf</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="32_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/>

<a NAME="E1:32_1"/>
( <span class="p1">(<span class="default"><a href="fuzzy_3.html#K1">Zmf</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> 0 &amp; <span class="p1">(<span class="default"><a href="fuzzy_3.html#K1">Zmf</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> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="fuzzy_4.html#T20">FUZZY_4:20</a></i>;<br/>
<b>hence </b><a NAME="E2:32_1"/>
<span class="p1">(<span class="default"><a href="fuzzy_3.html#K1">Zmf</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_3.html#K1">Zmf</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>

<b>thus </b><a NAME="E2:32"/>
 <a href="fuzzy_3.html#K1">Zmf</a> <font color="Maroon">X</font>,<font color="Maroon">X</font> is <a href="lfuzzy_1.html#V4" target="_self">antisymmetric</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">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#D9" target="_self">LFUZZY_1:def 9</a><br/>

<b>thus </b><a NAME="E1:32_2"/>
( <span class="p1">(<span class="default"><a href="fuzzy_3.html#K1">Zmf</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#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> implies <span class="p1">(<span class="default"><a href="fuzzy_3.html#K1">Zmf</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> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="fuzzy_4.html#T20">FUZZY_4:20</a></i>;<br/>


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

<b>thus </b><a NAME="E3:32"/>
 <a href="fuzzy_3.html#K1">Zmf</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="32_3"><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/>

<i><font color="Green">E49</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_3.html#K1">Zmf</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_3.html#K1">Zmf</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_3.html#K1">Zmf</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_3.html#K1">Zmf</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> 0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_3.html#K1">Zmf</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="fuzzy_4.html#T20">FUZZY_4:20</a></i>
<br/>.= 
 <a href="xxreal_0.html#K3">min</a> 0,0
<b>by </b><i><a class="ref" href="fuzzy_4.html#T20">FUZZY_4:20</a></i>
<br/>.= 
0

;<br/>
<a NAME="E2:32_3"/>
<span class="p1">(<span class="default"><a href="fuzzy_3.html#K1">Zmf</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> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="fuzzy_4.html#T20">FUZZY_4:20</a></i>;<br/>
<b>hence </b><a NAME="E3:32_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_3.html#K1">Zmf</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_3.html#K1">Zmf</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_3.html#K1">Zmf</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>
