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

<span class="kw">thus </span><a NAME="E1:17_1_1"/>
( <font color="Maroon" title="c2">R</font> is  <a href="lfuzzy_1.html#V3" title="LFUZZY_1:attr.3">transitive</a>  iff for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</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"><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="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</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="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&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="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><span class="kw">hereby </span> <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">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</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"><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="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</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="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&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="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) implies <font color="Maroon" title="c2">R</font> is  <a href="lfuzzy_1.html#V3" title="LFUZZY_1:attr.3">transitive</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:17_1_1_1_1"/>
<font color="Maroon" title="c2">R</font> is  <a href="lfuzzy_1.html#V3" title="LFUZZY_1:attr.3">transitive</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">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</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"><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="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</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="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&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="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/><a NAME="E2:17_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<font color="Maroon" title="c2">R</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lfuzzy_1.html#D6" target="_self" title="LFUZZY_1:def.6">Def6</a></span>;<br/><span class="kw">thus </span><a NAME="E3:17_1_1_1_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</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"><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="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</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="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&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="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</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">y</font>,<font color="Maroon" title="c5">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&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="c3">x</font>,<font color="Maroon" title="c5">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/>

<a NAME="E1:17_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">y9</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Olive" title="b1">y9</font>,<font color="Maroon" title="c5">z</font></span>)</span></span>)</span> where <font color="Olive" title="b1">y9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</span> 
 
;<br/>
<a NAME="E2:17_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span>)</span> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&lt;=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">y9</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Olive" title="b1">y9</font>,<font color="Maroon" title="c5">z</font></span>)</span></span>)</span> where <font color="Olive" title="b1">y9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T24" title="YELLOW_2:th.24">YELLOW_2:24</a></span>;<br/>
<a NAME="E3:17_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span>)</span> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&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#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lfuzzy_0.html#T22" title="LFUZZY_0:th.22">LFUZZY_0:22</a></span>;<br/>
<a NAME="E4:17_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font></span>)</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#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lfuzzy_0.html#T3" title="LFUZZY_0:th.3">LFUZZY_0:3</a></span>;<br/>
<a NAME="E5:17_1_1_1_1_1"/>
<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="c3">x</font>,<font color="Maroon" title="c5">z</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="c3">x</font>,<font color="Maroon" title="c5">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1_1_1_1"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="lfuzzy_1.html#D1" target="_self" title="LFUZZY_1:def.1">Def1</a></span>;<br/>
<a NAME="E6:17_1_1_1_1_1"/><span class="kw">then </span>
<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="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</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">y</font>,<font color="Maroon" title="c5">z</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> <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="c3">x</font>,<font color="Maroon" title="c5">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1_1_1_1_1"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:17_1_1_1_1_1"/>
<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="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</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">y</font>,<font color="Maroon" title="c5">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&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="c3">x</font>,<font color="Maroon" title="c5">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lfuzzy_0.html#T3" title="LFUZZY_0:th.3">LFUZZY_0:3</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/>


</div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:17_1_1_1"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</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"><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="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</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="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&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="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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> </span><br/>

<span class="kw">thus </span><a NAME="E3:17_1_1_1"/>
<font color="Maroon" title="c2">R</font> <a href="fuzzy_1.html#NR2" title="FUZZY_1:NR.2">c=</a> 
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lfuzzy_1.html#D6" target="_self" title="LFUZZY_1:def.6">LFUZZY_1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="c3">x</font>,<font color="Maroon" title="c4">z</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="c3">x</font>,<font color="Maroon" title="c4">z</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">W</font> =  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">z</font></span>)</span>;<br/>
<a NAME="E1:17_1_1_1_2"/>
for <font color="Olive" title="b1">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">z</font></span>)</span> holds <br/><font color="Olive" title="b1">r</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#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="c3">x</font>,<font color="Maroon" title="c4">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">r</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">z</font></span>)</span> implies <font color="Maroon" title="c6">r</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#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="c3">x</font>,<font color="Maroon" title="c4">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:17_1_1_1_2_1"/>
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">z</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">r</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#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="c3">x</font>,<font color="Maroon" title="c4">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/>

<a NAME="E2:17_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">y9</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y9</font>,<font color="Maroon" title="c4">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">y9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : verum </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="lfuzzy_1.html#E15"><span class="lab"><font color="Green" title="E12">Lm1</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E4:17_1_1_1_2_1"/><span class="lab"><font color="Green" title="E14">A4</font></span>: 
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">x</font>,<font color="Maroon" title="c7">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</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="c7">y</font>,<font color="Maroon" title="c4">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>
 ;<br/>
<a NAME="E5:17_1_1_1_2_1"/>
<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="c3">x</font>,<font color="Maroon" title="c7">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</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="c7">y</font>,<font color="Maroon" title="c4">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&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="c3">x</font>,<font color="Maroon" title="c4">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1_1_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:17_1_1_1_2_1"/>
<font color="Maroon" title="c6">r</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#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="c3">x</font>,<font color="Maroon" title="c4">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1_1_1_2_1"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="lfuzzy_0.html#T3" title="LFUZZY_0:th.3">LFUZZY_0:3</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/>


</div><span class="kw">end;</span></div>
<a NAME="E2:17_1_1_1_2"/><span class="kw">then </span>
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">z</font></span>)</span></span>)</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="c3">x</font>,<font color="Maroon" title="c4">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr3.html#T1" title="TOPMETR3:th.1">TOPMETR3:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:17_1_1_1_2"/>
<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="c3">x</font>,<font color="Maroon" title="c4">z</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="c3">x</font>,<font color="Maroon" title="c4">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fuzzy_4.html#D3" title="FUZZY_4:def.3">FUZZY_4:def 3</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/>


</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>


</div>
