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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>let </b><font color="Maroon">h</font> be    <a href="fuzzy_1.html#M1">Membership_Func</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">h1</font> be    <a href="fuzzy_1.html#M1">Membership_Func</a> of <font color="Maroon">C</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:7"/><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>
 <b>and </b><br/><a NAME="E2:7"/><i><font color="Green">E28</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">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h1</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="E3:7"/>
( <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> &amp; <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:7"/><b>then </b>
( <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> &amp; 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/>
<a NAME="E5:7"/><b>then </b>
 <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">c</font></span>)</span>,<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <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">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T18">XXREAL_0:18</a></i>;<br/>
<a NAME="E6:7"/><b>then </b>
 <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">c</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">h1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <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">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</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 NAME="E7:7"/><b>then </b>
 <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">c</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">h1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <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#K3">1_minus</a> <font color="Maroon">h</font></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#D6">FUZZY_1:def 6</a></i>;<br/>
<a NAME="E8:7"/><b>then </b>
<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">h1</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>  <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#K3">1_minus</a> <font color="Maroon">h</font></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 NAME="E9:7"/><b>then </b>
<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">h1</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> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K3">1_minus</a> <font color="Maroon">h</font></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#T6">FUZZY_1:6</a></i>;<br/>
<a NAME="E10:7"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="fuzzy_2.html#K1" target="_self">\</a> <font color="Maroon">h1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K1">min</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K3">1_minus</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font>
 
;<br/>
<b>hence </b><a NAME="E11:7"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="fuzzy_2.html#K1" target="_self">\</a> <font color="Maroon">h1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="fuzzy_2.html#K1" target="_self">\</a> <font color="Maroon">h</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font>
 ;<br/>


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