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

<b>let </b><font color="Maroon">C</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">f</font> be    <a href="fuzzy_1.html#M1" target="_self">Membership_Func</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">g</font> be    <a href="fuzzy_1.html#M1" target="_self">Membership_Func</a> of <font color="Maroon">C</font>;<br/>





<b>thus </b><a NAME="E1:38"/>
(  <a href="fuzzy_1.html#K2" target="_self">max</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font> implies ( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_1"/><i><font color="Green">E23</font></i>: 
 <a href="fuzzy_1.html#K2" target="_self">max</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font>
 ;<br/>

<b>thus </b><a NAME="E2:38_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:38_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E2:38_1_1"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_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">C</font>;<br/>

<a NAME="E1:38_1_1_1"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:38_1_1_1"/>
 <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="fuzzy_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E3:38_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<b>hence </b><a NAME="E4:38_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> implies <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</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="E3:38_1_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="fuzzy_1.html#T2" target="_self">Th2</a>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


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

<b>thus </b><a NAME="E3:38_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:38_1_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E2:38_1_2"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> holds <br/><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2_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">C</font>;<br/>

<a NAME="E1:38_1_2_1"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:38_1_2_1"/>
 <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="fuzzy_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E3:38_1_2_1"/><b>then </b>
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<b>hence </b><a NAME="E4:38_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> implies <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</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="E3:38_1_2"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="fuzzy_1.html#T2" target="_self">Th2</a>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


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


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

<b>assume </b><a NAME="E2:38"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font> )
 ;<br/>

<a NAME="E3:38"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" target="_self">max</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> &amp; <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:38"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> holds <br/><span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" target="_self">max</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font>
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E5:38"/>
 <a href="fuzzy_1.html#K2" target="_self">max</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="fuzzy_1.html#T2" target="_self">Th2</a>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


</div>
