<?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>let </b><font color="Maroon" title="c2">R</font>, <font color="Maroon" title="c3">S</font>, <font color="Maroon" title="c4">T</font> be    <a href="fuzzy_1.html#M1" title="FUZZY_1:mode.1">Membership_Func</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>assume </b><a NAME="E1:6"/><i><font color="Green" title="E6">A1</font></i>: 
( <font color="Maroon" title="c3">S</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  &amp; <font color="Maroon" title="c4">T</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  )
 ;<br/>

<a NAME="E2:6"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">T</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>

<a NAME="E1:6_1"/>
( <font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">S</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c3">S</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">T</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Maroon" title="c5">x</font> )
 
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E1:6"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="fuzzy_1.html#D3" title="FUZZY_1:def.3">FUZZY_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E2:6_1"/>
<font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">T</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:6"/>
<font color="Maroon" title="c4">T</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a> 
 <b>by </b><i><a class="ref" href="fuzzy_1.html#D3" title="FUZZY_1:def.3">FUZZY_1:def 3</a></i>;<br/>


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