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<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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="fuzzy_1.html#M1" title="FUZZY_1:mode.1">Membership_Func</a> of <font color="Maroon" title="c1">C</font> holds  <span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font></span>)</span> <a href="fuzzy_2.html#K1" title="FUZZY_2:func.1">\</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font></span>)</span> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a> </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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="fuzzy_2.html#K1" title="FUZZY_2:func.1">\</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a> </span><br/><b>let </b><font color="Maroon" title="c4">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="fuzzy_1.html#D3" title="FUZZY_1:def.3">FUZZY_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="fuzzy_1.html#K6" title="FUZZY_1:func.6">\+\</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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="fuzzy_2.html#K1" title="FUZZY_2:func.1">\</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">c</font></span><br/>







<span class="p1">(<span class="default"><span class="p2">(<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="fuzzy_2.html#K1" title="FUZZY_2:func.1">\</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">c</font> = 
<span class="p1">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <span class="p2">(<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>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <span class="p3">(<span class="default"><a href="fuzzy_1.html#K3" title="FUZZY_1:func.3">1_minus</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="fuzzy_1.html#K3" title="FUZZY_1:func.3">1_minus</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">c</font>
<b>by </b><i><a class="ref" href="fuzzy_1.html#T12" title="FUZZY_1:th.12">FUZZY_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <span class="p3">(<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>,<span class="p3">(<span class="default"><a href="fuzzy_1.html#K3" title="FUZZY_1:func.3">1_minus</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <span class="p3">(<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>,<span class="p3">(<span class="default"><a href="fuzzy_1.html#K3" title="FUZZY_1:func.3">1_minus</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">c</font>
<b>by </b><i><a class="ref" href="fuzzy_1.html#T10" title="FUZZY_1:th.10">FUZZY_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <span class="p3">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K3" title="FUZZY_1:func.3">1_minus</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c2">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K3" title="FUZZY_1:func.3">1_minus</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c3">g</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <span class="p3">(<span class="default"><a href="fuzzy_1.html#K3" title="FUZZY_1:func.3">1_minus</a> <font color="Maroon" title="c3">g</font></span>)</span>,<span class="p3">(<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></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">c</font>
<b>by </b><i><a class="ref" href="fuzzy_1.html#T10" title="FUZZY_1:th.10">FUZZY_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <span class="p3">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K3" title="FUZZY_1:func.3">1_minus</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c2">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K3" title="FUZZY_1:func.3">1_minus</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c3">g</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <span class="p3">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K3" title="FUZZY_1:func.3">1_minus</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c2">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K3" title="FUZZY_1:func.3">1_minus</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c3">g</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">c</font>
<b>by </b><i><a class="ref" href="fuzzy_1.html#T10" title="FUZZY_1:th.10">FUZZY_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <span class="p3">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <span class="p5">(<span class="default"><a href="fuzzy_1.html#K3" title="FUZZY_1:func.3">1_minus</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c2">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <span class="p5">(<span class="default"><a href="fuzzy_1.html#K3" title="FUZZY_1:func.3">1_minus</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c3">g</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K3" title="FUZZY_1:func.3">1_minus</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <span class="p3">(<span class="default"><a href="fuzzy_1.html#K3" title="FUZZY_1:func.3">1_minus</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">c</font>
<b>by </b><i><a class="ref" href="fuzzy_1.html#T8" title="FUZZY_1:th.8">FUZZY_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <span class="p3">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K3" title="FUZZY_1:func.3">1_minus</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c2">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <span class="p5">(<span class="default"><a href="fuzzy_1.html#K3" title="FUZZY_1:func.3">1_minus</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c3">g</font></span>)</span>,<span class="p4">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <span class="p5">(<span class="default"><a href="fuzzy_1.html#K3" title="FUZZY_1:func.3">1_minus</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <span class="p3">(<span class="default"><a href="fuzzy_1.html#K3" title="FUZZY_1:func.3">1_minus</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">c</font>
<b>by </b><i><a class="ref" href="fuzzy_1.html#T8" title="FUZZY_1:th.8">FUZZY_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <span class="p3">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K3" title="FUZZY_1:func.3">1_minus</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c3">g</font></span>)</span>,<span class="p3">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K3" title="FUZZY_1:func.3">1_minus</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <span class="p3">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K3" title="FUZZY_1:func.3">1_minus</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c2">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K3" title="FUZZY_1:func.3">1_minus</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c3">g</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">c</font>
<b>by </b><i><a class="ref" href="fuzzy_1.html#T8" title="FUZZY_1:th.8">FUZZY_1:8</a></i>
<br/>.= 
 <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="fuzzy_1.html#K6" title="FUZZY_1:func.6">\+\</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">c</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <span class="p3">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K3" title="FUZZY_1:func.3">1_minus</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c2">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <span class="p4">(<span class="default"><a href="fuzzy_1.html#K3" title="FUZZY_1:func.3">1_minus</a> <font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">c</font></span>)</span>
<b>by </b><i><a class="ref" href="fuzzy_1.html#T6" title="FUZZY_1:th.6">FUZZY_1:6</a></i>
;<br/>
<b>hence </b><a NAME="E2:18"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="fuzzy_1.html#K6" title="FUZZY_1:func.6">\+\</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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="fuzzy_2.html#K1" title="FUZZY_2:func.1">\</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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