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

<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font>, <font color="Maroon" title="c3">Z</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> ;<br/><b>let </b><font color="Maroon" title="c4">R</font> be   <a href="fuzzy_3.html#NM1" title="FUZZY_3:NM.1">RMembership_Func</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>;<br/><b>let </b><font color="Maroon" title="c5">S</font> be   <a href="fuzzy_3.html#NM1" title="FUZZY_3:NM.1">RMembership_Func</a> of <font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c3">Z</font>;<br/><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c7">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">Z</font>;<br/>









<a NAME="E1:59"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">Z</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
;<br/>
<a NAME="E2:59"/><b>then </b><i><font color="Green" title="E26">A1</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c5">S</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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#K5" title="RELSET_1:func.5">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="c4">R</font>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">z</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="fuzzy_4.html#D3" title="FUZZY_4:def.3">FUZZY_4:def 3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">L</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Olive" title="b1">y</font>,<font color="Maroon" title="c7">z</font></span>)</span></span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : verum </span>}</span> ;<br/>
<a NAME="E3:59"/><i><font color="Green" title="E27">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c5">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="c6">x</font>,<font color="Maroon" title="c7">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Olive" title="b1">y</font>,<font color="Maroon" title="c7">z</font></span>)</span></span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<a NAME="E1:59_1"/>
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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>  st <font color="Olive" title="b1">b</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="c4">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c6">x</font>,<font color="Olive" title="b2">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Olive" title="b2">y</font>,<font color="Maroon" title="c7">z</font></span>)</span></span>)</span> where <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : verum </span>}</span>  holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c5">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="c6">x</font>,<font color="Maroon" title="c7">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="lfuzzy_0.html#NR2" title="LFUZZY_0:NR.2">&gt;&gt;=</a> <font color="Olive" title="b1">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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/>

<b>assume </b><a NAME="E1:59_1_1"/>
<font color="Maroon" title="c9">b</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="c4">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Olive" title="b1">y</font>,<font color="Maroon" title="c7">z</font></span>)</span></span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font><b> such that </b><br/><a NAME="E3:59_1_1"/><i><font color="Green" title="E28">A3</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c10">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c7">z</font></span>)</span>
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c11">b</font> = <font color="Maroon" title="c9">b</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="lfuzzy_0.html#E31"><i><font color="Green" title="E23">Lm4</font></i></a></i>;<br/>
<a NAME="E5:59_1_1"/><i><font color="Green" title="E29">A4</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D1" title="FUZZY_1:def.1">FUZZY_1:def 1</a></i>;<br/>
<a NAME="E6:59_1_1"/>
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">z</font></span>)</span> <a href="lfuzzy_0.html#K7" title="LFUZZY_0:func.7">.</a> <font color="Maroon" title="c10">y</font>
 
<b>by </b><i><a class="txt" href="lfuzzy_0.html#E3:59_1_1"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="fuzzy_4.html#D2" title="FUZZY_4:def.2">FUZZY_4:def 2</a></i>;<br/>
<a NAME="E7:59_1_1"/><b>then </b>
<font color="Maroon" title="c11">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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#K5" title="RELSET_1:func.5">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="c4">R</font>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">z</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="lfuzzy_0.html#E5:59_1_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="fuzzy_4.html#T1" title="FUZZY_4:th.1">FUZZY_4:1</a></i>;<br/>
<b>hence </b><a NAME="E8:59_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c5">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="c6">x</font>,<font color="Maroon" title="c7">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="lfuzzy_0.html#NR2" title="LFUZZY_0:NR.2">&gt;&gt;=</a> <font color="Maroon" title="c9">b</font>
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E2:59"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="lfuzzy_0.html#T3" target="_self" title="LFUZZY_0:th.3">Th3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:59_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c5">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="c6">x</font>,<font color="Maroon" title="c7">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Olive" title="b1">y</font>,<font color="Maroon" title="c7">z</font></span>)</span></span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:59"/>
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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>  st <font color="Olive" title="b1">b</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c6">x</font>,<font color="Olive" title="b2">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Olive" title="b2">y</font>,<font color="Maroon" title="c7">z</font></span>)</span></span>)</span> where <font color="Olive" title="b2">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : verum </span>}</span>  holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c5">S</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">z</font> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&lt;=</a> <font color="Olive" title="b1">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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/>

<b>assume </b><a NAME="E1:59_2"/><i><font color="Green" title="E28">A5</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Olive" title="b1">y</font>,<font color="Maroon" title="c7">z</font></span>)</span></span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : verum </span>}</span> 
 ;<br/>

<a NAME="E2:59_2"/><i><font color="Green" title="E29">A6</font></i>: 
( <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">z</font></span>)</span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">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="c4">R</font>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">z</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> &amp;  <a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">z</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">z</font></span>)</span>, <a href="relset_1.html#K5" title="RELSET_1:func.5">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="c4">R</font>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">z</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D1" title="FUZZY_1:def.1">FUZZY_1:def 1</a>, <a class="ref" href="partfun1.html#T24" title="PARTFUN1:th.24">PARTFUN1:24</a></i>;<br/>
<a NAME="E3:59_2"/><i><font color="Green" title="E30">A7</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Olive" title="b1">y</font>,<font color="Maroon" title="c7">z</font></span>)</span></span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : verum </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">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="c4">R</font>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">z</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lfuzzy_0.html#E32"><i><font color="Green" title="E24">Lm5</font></i></a></i>;<br/>
<a NAME="E4:59_2"/><i><font color="Green" title="E31">A8</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">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="c4">R</font>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">z</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="lfuzzy_0.html#E32"><i><font color="Green" title="E24">Lm5</font></i></a></i>;<br/>
<a NAME="E5:59_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#K5" title="RELSET_1:func.5">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="c4">R</font>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">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="c9">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">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> ;<br/>

<b>assume </b><a NAME="E1:59_2_1"/><i><font color="Green" title="E32">A9</font></i>: 
<font color="Maroon" title="c10">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">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="c4">R</font>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">z</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c11">r</font> = <font color="Maroon" title="c10">r</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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> <b>by </b><i><a class="txt" href="lfuzzy_0.html#E2:59_2"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="lfuzzy_0.html#D3" target="_self" title="LFUZZY_0:def.3">Def3</a></i>;<br/>
<a NAME="E3:59_2_1"/>
<font color="Maroon" title="c11">r</font> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&lt;=</a> <font color="Maroon" title="c9">b</font>
 
<b>by </b><i><a class="txt" href="lfuzzy_0.html#E1:59_2"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="lfuzzy_0.html#E3:59_2"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="lfuzzy_0.html#E1:59_2_1"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E4:59_2_1"/>
<font color="Maroon" title="c10">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">b</font>
 <b>by </b><i><a class="ref" href="lfuzzy_0.html#T3" target="_self" title="LFUZZY_0:th.3">Th3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:59_2"/><b>then </b>
 <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#K5" title="RELSET_1:func.5">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="c4">R</font>,<font color="Maroon" title="c5">S</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">z</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">b</font>
 
<b>by </b><i><a class="txt" href="lfuzzy_0.html#E4:59_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="topmetr3.html#T1" title="TOPMETR3:th.1">TOPMETR3:1</a></i>;<br/>
<b>hence </b><a NAME="E7:59_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c5">S</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">z</font> <a href="lfuzzy_0.html#NR1" title="LFUZZY_0:NR.1">&lt;&lt;=</a> <font color="Maroon" title="c9">b</font>
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E2:59"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="lfuzzy_0.html#T3" target="_self" title="LFUZZY_0:th.3">Th3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:59"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c5">S</font></span>)</span> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">R</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">y</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="lfuzzy_0.html#K8" title="LFUZZY_0:func.8">.</a> <font color="Olive" title="b1">y</font>,<font color="Maroon" title="c7">z</font></span>)</span></span>)</span> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</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>
 <b>by </b><i><a class="txt" href="lfuzzy_0.html#E3:59"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="yellow_0.html#T32" title="YELLOW_0:th.32">YELLOW_0:32</a></i>;<br/>


</div>
