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<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="E1:25"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font> holds  <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">c</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="fuzzy_1.html#D3">FUZZY_1:def 3</a><br/>

<b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="fuzzy_1.html#D3">FUZZY_1:def 3</a><br/>

<a NAME="E2:25"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font>
 
<b>by </b><i/>;<br/>
<span class="p1">(<span class="default"><a href="fuzzy_1.html#K2">max</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="fuzzy_2.html#K1" target="_self">\</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> = 
<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#K2">max</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K2">max</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="fuzzy_1.html#K3">1_minus</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="fuzzy_1.html#T10">FUZZY_1:10</a></i>
<br/>.= 
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_1.html#K2">max</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_1.html#K2">max</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="fuzzy_1.html#K3">1_minus</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</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"><a href="square_1.html#K2">max</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_1.html#K2">max</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="fuzzy_1.html#K3">1_minus</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</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">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_1.html#K2">max</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="fuzzy_1.html#K3">1_minus</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>
;<br/>
<b>hence </b><a NAME="E4:25"/>
<span class="p1">(<span class="default"><a href="fuzzy_1.html#K2">max</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="fuzzy_2.html#K1" target="_self">\</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/>


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