<?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">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">Membership_Func</a> of <font color="Maroon">C</font>;<br/>





<b>assume </b><a NAME="E1:28"/><i><font color="Green">E26</font></i>: 
 <a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="fuzzy_1.html#K4">EMF</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E2:28"/>
 <a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="fuzzy_1.html#K3">1_minus</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<a NAME="E1:28_1"/><i><font color="Green">E28</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">min</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K3">1_minus</a> <font color="Maroon">g</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><a class="ref" href="fuzzy_1.html#D1">FUZZY_1:def 1</a></i>;<br/>
<a NAME="E2:28_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">min</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K3">1_minus</a> <font color="Maroon">g</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="28_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="E1:28_1_1"/><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><a href="fuzzy_1.html#K4">EMF</a> <font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <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"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="fuzzy_1.html#T6">FUZZY_1:6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_1_1_1"/>
( <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">EMF</a> <font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> or <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">EMF</a> <font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> )
 </span><b>by </b><i>, <a class="ref" href="xxreal_0.html#T15">XXREAL_0:15</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:28_1_1_1_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="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K4">EMF</a> <font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:28_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K3">1_minus</a> <font color="Maroon">g</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<span class="p1">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K3">1_minus</a> <font color="Maroon">g</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#K3">1_minus</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="fuzzy_1.html#T6">FUZZY_1:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K3">1_minus</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K4">EMF</a> <font color="Maroon">C</font></span>)</span></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#T6">FUZZY_1:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fuzzy_1.html#K4">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#T19">FUZZY_1:19</a></i>
;<br/>
<b>hence </b><a NAME="E2:28_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K3">1_minus</a> <font color="Maroon">g</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>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:28_1_1_1_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">min</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K3">1_minus</a> <font color="Maroon">g</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:28_1_1_1_1_2"/><i><font color="Green">E35</font></i>: 
<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">EMF</a> <font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:28_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K3">1_minus</a> <font color="Maroon">g</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<span class="p1">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K3">1_minus</a> <font color="Maroon">g</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#K3">1_minus</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="fuzzy_1.html#T6">FUZZY_1:6</a></i>
<br/>.= 
 <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">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fuzzy_1.html#K4">EMF</a> <font color="Maroon">C</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="fuzzy_1.html#D6">FUZZY_1:def 6</a></i>
<br/>.= 
 <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#K3">1_minus</a> <span class="p3">(<span class="default"><a href="fuzzy_1.html#K4">EMF</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="fuzzy_1.html#D6">FUZZY_1:def 6</a></i>
<br/>.= 
 <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">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><a class="ref" href="fuzzy_1.html#T44">FUZZY_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K5">UMF</a> <font color="Maroon">C</font></span>)</span></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#T6">FUZZY_1:6</a></i>
;<br/>
<b>hence </b><a NAME="E2:28_1_1_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K3">1_minus</a> <font color="Maroon">g</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>
 <b>by </b><i><a class="ref" href="fuzzy_1.html#T19">FUZZY_1:19</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:28_1_1_1_1_2"/>
( <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">min</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K3">1_minus</a> <font color="Maroon">g</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></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:28_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">min</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K3">1_minus</a> <font color="Maroon">g</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="E3:28_1"/>
 <a href="fuzzy_1.html#K1">min</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="fuzzy_1.html#K3">1_minus</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:28"/>
<font color="Maroon">f</font> <a href="fuzzy_2.html#K1" target="_self">\</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 ;<br/>


</div>
