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<div class="add">

<span class="kw">let </span><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#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#NM1" title="FUZZY_1:NM.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/><span class="kw">let </span><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</font> be   <a href="fuzzy_1.html#NM1" title="FUZZY_1:NM.1">Membership_Func</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="fuzzy_1.html#D2" onmouseover="rs('fuzzy_1/D2')" onmouseout="rh()">FUZZY_1:def 2</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="funct_2.html#K3" title="FUNCT_2:func.3">.</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="funct_2.html#K3" title="FUNCT_2:func.3">.</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="funct_2.html#K3" title="FUNCT_2:func.3">.</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">c</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fuzzy_1.html#T11" onmouseover="rs('fuzzy_1/T11')" onmouseout="rh()">FUZZY_1:11</a></span>
<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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">c</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fuzzy_1.html#T9" onmouseover="rs('fuzzy_1/T9')" onmouseout="rh()">FUZZY_1:9</a></span>
<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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">c</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fuzzy_1.html#T9" onmouseover="rs('fuzzy_1/T9')" onmouseout="rh()">FUZZY_1:9</a></span>
<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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">c</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fuzzy_1.html#T9" onmouseover="rs('fuzzy_1/T9')" onmouseout="rh()">FUZZY_1:9</a></span>
<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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">c</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fuzzy_1.html#T7" onmouseover="rs('fuzzy_1/T7')" onmouseout="rh()">FUZZY_1:7</a></span>
<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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">c</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fuzzy_1.html#T7" onmouseover="rs('fuzzy_1/T7')" onmouseout="rh()">FUZZY_1:7</a></span>
<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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">c</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fuzzy_1.html#T7" onmouseover="rs('fuzzy_1/T7')" onmouseout="rh()">FUZZY_1:7</a></span>
<br/>.= 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">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="funct_2.html#K3" title="FUNCT_2:func.3">.</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">c</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="fuzzy_1.html#T5" onmouseover="rs('fuzzy_1/T5')" onmouseout="rh()">FUZZY_1:5</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:17"/>
<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="funct_2.html#K3" title="FUNCT_2:func.3">.</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" onmouseover="rs('xxreal_0/T25')" onmouseout="rh()">XXREAL_0:25</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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