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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>, <font color="Olive" title="b3">h</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   <a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <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="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font></span>)</span>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b3">h</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <font color="Olive" title="b3">h</font>,<font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <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> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font></span>)</span>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b3">h</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Olive" title="b3">h</font>,<font color="Olive" title="b1">f</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</font>, <font color="Maroon" title="c4">h</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">  <a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <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> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <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> <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#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c2">f</font></span>)</span></span><br/>







<b>thus </b> <a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <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> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c2">f</font></span>)</span> = 
 <a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <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#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#K2" title="FUZZY_1:func.2">max</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">h</font></span>)</span>,<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#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#K2" title="FUZZY_1:func.2">max</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">f</font></span>)</span>
<b>by </b><i><a class="ref" href="fuzzy_1.html#T10" target="_self" title="FUZZY_1:th.10">Th10</a></i>
<br/>.= 
 <a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <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#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="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span>,<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#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#K2" title="FUZZY_1:func.2">max</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">f</font></span>)</span>
<b>by </b><i><a class="ref" href="fuzzy_1.html#T8" target="_self" title="FUZZY_1:th.8">Th8</a></i>
<br/>.= 
 <a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <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#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="c4">h</font>,<font color="Maroon" title="c3">g</font></span>)</span>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span>,<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#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>,<font color="Maroon" title="c2">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fuzzy_1.html#T8" target="_self" title="FUZZY_1:th.8">Th8</a></i>
<br/>.= 
 <a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <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>,<font color="Maroon" title="c4">h</font></span>)</span>,<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#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>,<font color="Maroon" title="c2">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fuzzy_1.html#T9" target="_self" title="FUZZY_1:th.9">Th9</a></i>
<br/>.= 
 <a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <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>,<font color="Maroon" title="c4">h</font></span>)</span>,<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>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fuzzy_1.html#T9" target="_self" title="FUZZY_1:th.9">Th9</a></i>
<br/>.= 
 <a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">h</font>,<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>)</span>,<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> <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#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fuzzy_1.html#T10" target="_self" title="FUZZY_1:th.10">Th10</a></i>
<br/>.= 
 <a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <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> <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#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span>,<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> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c2">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c3">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fuzzy_1.html#T10" target="_self" title="FUZZY_1:th.10">Th10</a></i>
<br/>.= 
 <a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <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> <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#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span>,<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="c4">h</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c3">g</font></span>)</span>
<b>by </b><i><a class="ref" href="fuzzy_1.html#T8" target="_self" title="FUZZY_1:th.8">Th8</a></i>
<br/>.= 
 <a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <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> <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#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">h</font></span>)</span>,<span class="p3">(<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="c4">h</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c3">g</font></span>)</span>
<b>by </b><i><a class="ref" href="fuzzy_1.html#T8" target="_self" title="FUZZY_1:th.8">Th8</a></i>
<br/>.= 
 <a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <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> <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#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">h</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="fuzzy_1.html#K1" title="FUZZY_1:func.1">min</a> <font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c2">f</font></span>)</span>
<b>by </b><i><a class="ref" href="fuzzy_1.html#T8" target="_self" title="FUZZY_1:th.8">Th8</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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