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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E1:10_1_1"/><i><font color="Green">E8</font></i>: 
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 <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 
;<br/>
<a NAME="E2:10_1_1"/><i><font color="Green">E9</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font>
 
;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:10_1_1"/><i><font color="Green">E10</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] ) ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="partfun1.html#S2">PARTFUN1:sch 2</a>(<a class="txt" href="fuzzy_1.html#E1:10_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="fuzzy_1.html#E2:10_1_1"><i><font color="Green">E9</font></i></a>);<br/></i>
<a NAME="E5:10_1_1"/><i><font color="Green">E11</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_1"/><i><font color="Green">E12</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:10_1_1_1_1_1_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:10_1_1_1_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E3:10_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="fuzzy_1.html#E4:10_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="fuzzy_1.html#E1:10_1_1_1_1_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:10_1_1_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:10_1_1_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:10_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_1_2"/><i><font color="Green">E13</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> )
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E2:10_1_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:10_1_1_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</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="E3:10_1_1_1_2_1"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> as  <a href="integra1.html#V1">closed-interval</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="integra1.html#D1">INTEGRA1:def 1</a></i>;<br/>
<a NAME="E5:10_1_1_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/>
<a NAME="E6:10_1_1_1_2_1"/><b>then </b><i><font color="Green">E15</font></i>: 
( 0 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>7</sub></font> &amp; 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="integra1.html#T6">INTEGRA1:6</a></i>;<br/>
<a NAME="E7:10_1_1_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E8:10_1_1_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E9:10_1_1_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E10:10_1_1_1_2_1"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="fuzzy_1.html#E1:10_1_1_1_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="fuzzy_1.html#E6:10_1_1_1_2_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>
<a NAME="E11:10_1_1_1_2_1"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp;  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="square_1.html#T50">SQUARE_1:50</a></i>;<br/>
<a NAME="E12:10_1_1_1_2_1"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="fuzzy_1.html#E4:10_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="fuzzy_1.html#E3:10_1_1_1_2_1"><i><font color="Green">E14</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:10_1_1_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="fuzzy_1.html#E3:10_1_1_1_2_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="fuzzy_1.html#E6:10_1_1_1_2_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:10_1_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:10_1_1_1"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> )
 <b>by </b><i><a class="txt" href="fuzzy_1.html#E1:10_1_1_1"><i><font color="Green">E12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:10_1_1"/><b>then </b><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is    <a href="fuzzy_1.html#M1" target="_self">Membership_Func</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E7:10_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="fuzzy_1.html#E4:10_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="fuzzy_1.html#E5:10_1_1"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:10_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="fuzzy_1.html#M1" target="_self">Membership_Func</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="fuzzy_1.html#E6:10_1_1"><i><font color="Green">E12</font></i></a></i>;<br/>


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