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<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#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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   <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">f</font> <a href="lfuzzy_0.html#K6" title="LFUZZY_0:func.6">@</a> </span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font>,<font color="Olive" title="b2">g</font> <a href="lfuzzy_0.html#K6" title="LFUZZY_0:func.6">@</a> </span>)</span></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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">f</font> <a href="lfuzzy_0.html#K6" title="LFUZZY_0:func.6">@</a> </span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">g</font> <a href="lfuzzy_0.html#K6" title="LFUZZY_0:func.6">@</a> </span>)</span></span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">fg</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">f</font> <a href="lfuzzy_0.html#K6" title="LFUZZY_0:func.6">@</a> </span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">g</font> <a href="lfuzzy_0.html#K6" title="LFUZZY_0:func.6">@</a> </span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">R</font> =  <a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">J</font> = <font color="Maroon" title="c1">C</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>;<br/>
<a NAME="E1:50"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#D5" title="YELLOW_1:def.5">YELLOW_1:def 5</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">f</font> <a href="lfuzzy_0.html#K6" title="LFUZZY_0:func.6">@</a> </span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">g</font> <a href="lfuzzy_0.html#K6" title="LFUZZY_0:func.6">@</a> </span>)</span></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lfuzzy_0.html#K3" title="LFUZZY_0:func.3">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Maroon" title="c7">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Maroon" title="c7">c</font></span>)</span></span><br/><a NAME="E1:50_1"/>
( ( for <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span> <a href="waybel_3.html#K3" title="WAYBEL_3:func.3">.</a> <font color="Olive" title="b1">c</font> is   <a href="waybel_0.html#NM6" title="WAYBEL_0:NM.6">Semilattice</a> ) &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p3"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span> <a href="waybel_3.html#K3" title="WAYBEL_3:func.3">.</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lfuzzy_0.html#K1" title="LFUZZY_0:func.1">RealPoset</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="lfuzzy_0.html#K5" title="LFUZZY_0:func.5">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">f</font> <a href="lfuzzy_0.html#K6" title="LFUZZY_0:func.6">@</a> </span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">g</font> <a href="lfuzzy_0.html#K6" title="LFUZZY_0:func.6">@</a> </span>)</span></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Maroon" title="c7">c</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">f</font> <a href="lfuzzy_0.html#K6" title="LFUZZY_0:func.6">@</a> </span>)</span> <a href="lfuzzy_0.html#K9" title="LFUZZY_0:func.9">.</a> <font color="Maroon" title="c7">c</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">g</font> <a href="lfuzzy_0.html#K6" title="LFUZZY_0:func.6">@</a> </span>)</span> <a href="lfuzzy_0.html#K9" title="LFUZZY_0:func.9">.</a> <font color="Maroon" title="c7">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="lfuzzy_0.html#T11" target="_self" title="LFUZZY_0:th.11">Th11</a></span>
<br/>.= 
 <a href="lfuzzy_0.html#K3" title="LFUZZY_0:func.3">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Maroon" title="c7">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Maroon" title="c7">c</font></span>)</span>

;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:50"/>
 <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">f</font> <a href="lfuzzy_0.html#K6" title="LFUZZY_0:func.6">@</a> </span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">g</font> <a href="lfuzzy_0.html#K6" title="LFUZZY_0:func.6">@</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fuzzy_1.html#D4" title="FUZZY_1:def.4">FUZZY_1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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