<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">X</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">R</font>, <font color="Olive" title="b2">S</font> being  <a href="fuzzy_2.html#NM1" title="FUZZY_2:NM.1">RMembership_Func</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">R</font> is <a href="lfuzzy_1.html#V3" title="LFUZZY_1:attr.3">transitive</a> &amp; <font color="Olive" title="b2">S</font> is <a href="lfuzzy_1.html#V3" title="LFUZZY_1:attr.3">transitive</a> holds <br/> <a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Olive" title="b1">R</font>,<font color="Olive" title="b2">S</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a> </span><br/><b>let </b><font color="Maroon" title="c2">R</font>, <font color="Maroon" title="c3">S</font> be   <a href="fuzzy_2.html#NM1" title="FUZZY_2:NM.1">RMembership_Func</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">R</font> is <a href="lfuzzy_1.html#V3" title="LFUZZY_1:attr.3">transitive</a> &amp; <font color="Maroon" title="c3">S</font> is <a href="lfuzzy_1.html#V3" title="LFUZZY_1:attr.3">transitive</a> implies  <a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  )</span><br/>



<b>assume </b><a NAME="E1:24"/>
( <font color="Maroon" title="c2">R</font> is <a href="lfuzzy_1.html#V3" title="LFUZZY_1:attr.3">transitive</a> &amp; <font color="Maroon" title="c3">S</font> is <a href="lfuzzy_1.html#V3" title="LFUZZY_1:attr.3">transitive</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a> </span><br/>

<a NAME="E2:24"/><b>then </b><i><font color="Green" title="E17">A1</font></i>: 
( <font color="Maroon" title="c2">R</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  &amp; <font color="Maroon" title="c3">S</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  )
 
<b>by </b><i><a class="ref" href="lfuzzy_1.html#D6" target="_self" title="LFUZZY_1:def.6">Def6</a></i>;<br/>
<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lfuzzy_1.html#D1" target="_self" title="LFUZZY_1:def.1">LFUZZY_1:def 1</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">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font></span><br/><b>let </b><font color="Maroon" title="c5">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</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"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><br/>



<a NAME="E3:24"/>
 <a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a> 
 
<b>by </b><i><a class="ref" href="fuzzy_4.html#T15" title="FUZZY_4:th.15">FUZZY_4:15</a></i>;<br/>
<a NAME="E4:24"/><b>then </b><i><font color="Green" title="E18">A2</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D3" title="FUZZY_1:def.3">FUZZY_1:def 3</a></i>;<br/>
<a NAME="E5:24"/>
(  <a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  &amp;  <a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c3">S</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a>  )
 
<b>by </b><i><a class="ref" href="fuzzy_4.html#T16" title="FUZZY_4:th.16">FUZZY_4:16</a></i>;<br/>
<a NAME="E6:24"/><b>then </b>
( <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c3">S</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D3" title="FUZZY_1:def.3">FUZZY_1:def 3</a>, <a class="ref" href="fuzzy_1.html#D4" title="FUZZY_1:def.4">FUZZY_1:def 4</a></i>;<br/>
<a NAME="E7:24"/><b>then </b>
<span class="p1">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c3">S</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T18" title="XXREAL_0:th.18">XXREAL_0:18</a></i>;<br/>
<a NAME="E8:24"/><b>then </b><i><font color="Green" title="E19">A3</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c3">S</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:24"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:24"/>
( <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c3">S</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> )
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D4" title="FUZZY_1:def.4">FUZZY_1:def 4</a></i>;<br/>
<a NAME="E10:24"/><b>then </b>
( <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c3">S</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> )
 
<b>by </b><i><a class="txt" href="#E2:24"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="lfuzzy_1.html#D1" target="_self" title="LFUZZY_1:def.1">Def1</a>, <a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E11:24"/><b>then </b>
( <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c3">S</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">S</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:24"/><b>then </b>
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c3">S</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T18" title="XXREAL_0:th.18">XXREAL_0:18</a></i>;<br/>
<a NAME="E13:24"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E8:24"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E14:24"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font></span>)</span></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K1" title="LFUZZY_1:func.1">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>
 <b>by </b><i><a class="ref" href="fuzzy_1.html#D4" title="FUZZY_1:def.4">FUZZY_1:def 4</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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