<?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/>



<a NAME="E1:24"/>
(  <a href="fuzzy_1.html#K5" target="_self">UMF</a> <font color="Maroon">C</font> <a href="fuzzy_1.html#NR2" target="_self">c=</a>  &amp;  <a href="fuzzy_1.html#K2" target="_self">max</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="fuzzy_1.html#K5" target="_self">UMF</a> <font color="Maroon">C</font></span>)</span> <a href="fuzzy_1.html#NR2" target="_self">c=</a>  )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E2:24"/>
 <a href="fuzzy_1.html#K2" target="_self">max</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="fuzzy_1.html#K5" target="_self">UMF</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fuzzy_1.html#K5" target="_self">UMF</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>

<a NAME="E3:24"/><i><font color="Green">E23</font></i>: 
 <a href="fuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="fuzzy_1.html#K5" target="_self">UMF</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<a NAME="E1:24_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> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K5" target="_self">UMF</a> <font color="Maroon">C</font></span>)</span></span>)</span> &amp; <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E2:24_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/><span class="p1">(<span class="default"><a href="fuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K5" target="_self">UMF</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_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="E1:24_1_1"/><i><font color="Green">E34</font></i>: 
<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#K5" target="_self">UMF</a> <font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<span class="p1">(<span class="default"><a href="fuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K5" target="_self">UMF</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
 <a href="square_1.html#K1">min</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"><span class="p2">(<span class="default"><a href="fuzzy_1.html#K5" target="_self">UMF</a> <font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="fuzzy_1.html#E3:24"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>
;<br/>
<b>hence </b><a NAME="E3:24_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> implies <span class="p1">(<span class="default"><a href="fuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K5" target="_self">UMF</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:24_1"/>
 <a href="fuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="fuzzy_1.html#K5" target="_self">UMF</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="fuzzy_1.html#T1" target="_self">Th1</a>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:24"/><i><font color="Green">E41</font></i>: 
 <a href="fuzzy_1.html#K2" target="_self">max</a> <font color="Maroon">f</font>,<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="hidden.html#R1">=</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>proof </b></a><div class="add">

<a NAME="E1:24_2"/><i><font color="Green">E42</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>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font></span>)</span></span>)</span> &amp; <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E2:24_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/><span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" target="_self">max</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_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="E1:24_2_1"/><i><font color="Green">E43</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/>
<span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" target="_self">max</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
 <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"><span class="p2">(<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></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f</font> <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>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>
;<br/>
<b>hence </b><a NAME="E3:24_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> implies <span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" target="_self">max</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:24_2"/>
 <a href="fuzzy_1.html#K2" target="_self">max</a> <font color="Maroon">f</font>,<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="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="fuzzy_1.html#E3:24"><i><font color="Green">E23</font></i></a>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:24"/>
 <a href="fuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">f</font>,<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="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="24_3"><b>proof </b></a><div class="add">

<a NAME="E1:24_3"/><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#K1" target="_self">min</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font></span>)</span></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="E2:24_3"/>
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#K1" target="_self">min</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font></span>)</span></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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3_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="E1:24_3_1"/><i><font color="Green">E45</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/>
<span class="p1">(<span class="default"><a href="fuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
 <a href="square_1.html#K1">min</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"><span class="p2">(<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></span>)</span>
<b>by </b><i/>
<br/>.= 
<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>, <a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>
;<br/>
<b>hence </b><a NAME="E3:24_3_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> implies <span class="p1">(<span class="default"><a href="fuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">C</font></span>)</span></span>)</span> <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> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:24_3"/>
 <a href="fuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">f</font>,<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="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="txt" href="fuzzy_1.html#E3:24"><i><font color="Green">E23</font></i></a>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:24"/>
(  <a href="fuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="fuzzy_1.html#K5" target="_self">UMF</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp;  <a href="fuzzy_1.html#K2" target="_self">max</a> <font color="Maroon">f</font>,<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="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp;  <a href="fuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">f</font>,<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="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#T1" target="_self">Th1</a></i>;<br/>


</div>
