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

<b>let </b><font color="Maroon" title="c1">C</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">f</font>, <font color="Maroon" title="c3">g</font> be    <a href="fuzzy_1.html#M1" title="FUZZY_1:mode.1">Membership_Func</a> of <font color="Maroon" title="c1">C</font>;<br/>





<b>thus </b><a NAME="E1:38"/>
(  <a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fuzzy_1.html#K4" title="FUZZY_1:func.4">EMF</a> <font color="Maroon" title="c1">C</font> implies ( <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fuzzy_1.html#K4" title="FUZZY_1:func.4">EMF</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fuzzy_1.html#K4" title="FUZZY_1:func.4">EMF</a> <font color="Maroon" title="c1">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" title="E24">A1</font></i>: 
 <a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fuzzy_1.html#K4" title="FUZZY_1:func.4">EMF</a> <font color="Maroon" title="c1">C</font>
 ;<br/>

<b>thus </b><a NAME="E2:38_1"/>
<font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fuzzy_1.html#K4" title="FUZZY_1:func.4">EMF</a> <font color="Maroon" title="c1">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" title="E25">A2</font></i>: 
( <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" title="FUZZY_1:func.4">EMF</a> <font color="Maroon" title="c1">C</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D1" target="_self" title="FUZZY_1:def.1">Def1</a></i>;<br/>
<a NAME="E2:38_1_1"/>
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">C</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> holds <br/><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" title="FUZZY_1:func.4">EMF</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">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" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font>;<br/>

<a NAME="E1:38_1_1_1"/><i><font color="Green" title="E26">A3</font></i>: 
<span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" title="FUZZY_1:func.4">EMF</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#T17" target="_self" title="FUZZY_1:th.17">Th17</a></i>;<br/>
<a NAME="E2:38_1_1_1"/>
 <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" title="FUZZY_1:func.4">EMF</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D5" target="_self" title="FUZZY_1:def.5">Def5</a>, <a class="txt" href="fuzzy_1.html#E1:38_1"><i><font color="Green" title="E24">A1</font></i></a></i>;<br/>
<a NAME="E3:38_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" title="FUZZY_1:func.4">EMF</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/>
<b>hence </b><a NAME="E4:38_1_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" title="FUZZY_1:func.4">EMF</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font> )
 <b>by </b><i><a class="txt" href="fuzzy_1.html#E1:38_1_1_1"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:38_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fuzzy_1.html#K4" title="FUZZY_1:func.4">EMF</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="fuzzy_1.html#E1:38_1_1"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>;<br/>


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

<b>thus </b><a NAME="E3:38_1"/>
<font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fuzzy_1.html#K4" title="FUZZY_1:func.4">EMF</a> <font color="Maroon" title="c1">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" title="E25">A4</font></i>: 
( <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g</font> &amp; <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" title="FUZZY_1:func.4">EMF</a> <font color="Maroon" title="c1">C</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D1" target="_self" title="FUZZY_1:def.1">Def1</a></i>;<br/>
<a NAME="E2:38_1_2"/>
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">C</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> holds <br/><font color="Maroon" title="c3">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" title="FUZZY_1:func.4">EMF</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">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" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font>;<br/>

<a NAME="E1:38_1_2_1"/><i><font color="Green" title="E26">A5</font></i>: 
<span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" title="FUZZY_1:func.4">EMF</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#T17" target="_self" title="FUZZY_1:th.17">Th17</a></i>;<br/>
<a NAME="E2:38_1_2_1"/>
 <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" title="FUZZY_1:func.4">EMF</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D5" target="_self" title="FUZZY_1:def.5">Def5</a>, <a class="txt" href="fuzzy_1.html#E1:38_1"><i><font color="Green" title="E24">A1</font></i></a></i>;<br/>
<a NAME="E3:38_1_2_1"/><b>then </b>
<font color="Maroon" title="c3">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" title="FUZZY_1:func.4">EMF</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/>
<b>hence </b><a NAME="E4:38_1_2_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c3">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" title="FUZZY_1:func.4">EMF</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font> )
 <b>by </b><i><a class="txt" href="fuzzy_1.html#E1:38_1_2_1"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:38_1_2"/>
<font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fuzzy_1.html#K4" title="FUZZY_1:func.4">EMF</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="fuzzy_1.html#E1:38_1_2"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">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" title="E24">A6</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fuzzy_1.html#K4" title="FUZZY_1:func.4">EMF</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fuzzy_1.html#K4" title="FUZZY_1:func.4">EMF</a> <font color="Maroon" title="c1">C</font> )
 ;<br/>

<a NAME="E3:38"/><i><font color="Green" title="E25">A7</font></i>: 
( <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span> &amp; <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" title="FUZZY_1:func.4">EMF</a> <font color="Maroon" title="c1">C</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D1" target="_self" title="FUZZY_1:def.1">Def1</a></i>;<br/>
<a NAME="E4:38"/>
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">C</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> holds <br/><span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" title="FUZZY_1:func.4">EMF</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font>
 
<b>by </b><i><a class="txt" href="fuzzy_1.html#E2:38"><i><font color="Green" title="E24">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:38"/>
 <a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fuzzy_1.html#K4" title="FUZZY_1:func.4">EMF</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="fuzzy_1.html#E3:38"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>;<br/>


</div>
