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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="fuzzy_3.html#NM1">RMembership_Func</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>



<i><font color="Green">E20</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="fuzzy_1.html#D1">FUZZY_1:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="fuzzy_1.html#D1">FUZZY_1:def 1</a></i>
;<br/>
<a NAME="E2:39"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:39_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:39_2"/><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> ,<span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="lfuzzy_0.html#T22">LFUZZY_0:22</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> ;<br/>
<a NAME="E4:39_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:39_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:39_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:39_2_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
 ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_2_1_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:39_2_1_1_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><a NAME="E2:39_2_1_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="fuzzy_4.html#T4">FUZZY_4:4</a></i>;<br/><font color="Maroon">c<sub>8</sub></font> = 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E3:39_2_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="lfuzzy_0.html#T5">LFUZZY_0:5</a></i>
<br/>.= 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,1
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E1:39_2_1_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="fuzzy_4.html#D4">FUZZY_4:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E1:39_2_1_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="lfuzzy_1.html#E2:39_2_1_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>
;<br/><b>hence </b><a NAME="E4:39_2_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="lfuzzy_0.html#T3">LFUZZY_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:39_2_1_1_1_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><a NAME="E2:39_2_1_1_1_2"/><i><font color="Green">E25</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="fuzzy_4.html#T4">FUZZY_4:4</a></i>;<br/><font color="Maroon">c<sub>8</sub></font> = 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E3:39_2_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="lfuzzy_0.html#T5">LFUZZY_0:5</a></i>
<br/>.= 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,0
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E1:39_2_1_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="fuzzy_4.html#D4">FUZZY_4:def 4</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E2:39_2_1_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>
;<br/><a NAME="E4:39_2_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="fuzzy_4.html#T4">FUZZY_4:4</a></i>;<br/><b>hence </b><a NAME="E5:39_2_1_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="lfuzzy_0.html#T3">LFUZZY_0:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E2:39_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:39_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span>  holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lfuzzy_0.html#K1">RealPoset</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:39_2_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> 
 ;<br/>

<a NAME="E2:39_2_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="fuzzy_4.html#T4">FUZZY_4:4</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> = 
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="lfuzzy_0.html#T5">LFUZZY_0:5</a></i>
<br/>.= 
 <a href="xxreal_0.html#K3">min</a> 1,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="fuzzy_4.html#D4">FUZZY_4:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E2:39_2_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>
;<br/>
<a NAME="E4:39_2_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E5:39_2_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lfuzzy_0.html#K7">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="lfuzzy_1.html#E1:39_2_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:39_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="lfuzzy_1.html#E3:39_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="lfuzzy_1.html#E4:39_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:39"/>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K4">Imf</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="lfuzzy_1.html#E1:39"><i><font color="Green">E20</font></i></a>, <a class="ref" href="partfun1.html#T35">PARTFUN1:35</a></i>;<br/>


</div>
