<?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_1.html#M1">Membership_Func</a> of <font color="Maroon">X</font>, <font color="Maroon">S</font> be    <a href="fuzzy_1.html#M1">Membership_Func</a> of <font color="Maroon">X</font>;<br/>



<b>thus </b><a NAME="E1:5"/>
( <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> implies ( <font color="Maroon">S</font> <a href="fuzzy_1.html#NR2" target="_self">c=</a>  &amp; <font color="Maroon">R</font> <a href="fuzzy_1.html#NR2" target="_self">c=</a>  ) )
 ;<br/>

<b>assume </b><a NAME="E2:5"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">S</font> <a href="fuzzy_1.html#NR2" target="_self">c=</a>  &amp; <font color="Maroon">R</font> <a href="fuzzy_1.html#NR2" target="_self">c=</a>  )
 ;<br/>

<a NAME="E3:5"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds  <font color="Maroon">R</font> <a href="lfuzzy_0.html#K7">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="lfuzzy_0.html#K7">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_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>;<br/>

<a NAME="E1:5_1"/>
( <font color="Maroon">R</font> <a href="lfuzzy_0.html#K7">.</a> <font color="Maroon">x</font> <a href="lfuzzy_0.html#NR5" target="_self">&lt;R=</a> <font color="Maroon">S</font> <a href="lfuzzy_0.html#K7">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">S</font> <a href="lfuzzy_0.html#K7">.</a> <font color="Maroon">x</font> <a href="lfuzzy_0.html#NR5" target="_self">&lt;R=</a> <font color="Maroon">R</font> <a href="lfuzzy_0.html#K7">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, <a class="ref" href="fuzzy_1.html#D3">FUZZY_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E2:5_1"/>
<font color="Maroon">R</font> <a href="lfuzzy_0.html#K7">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="lfuzzy_0.html#K7">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:5"/>
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font>
 <b>by </b><i/>;<br/>


</div>
