<?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/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>





<b>assume </b><a NAME="E1:49"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:49"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <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> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K5" target="_self">TrCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <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>;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:49_1"/><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</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="txt" href="lfuzzy_1.html#E2:49_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E4:49_1"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="lfuzzy_1.html#K5" target="_self">TrCl</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>8</sub></font>,<font color="Maroon">c<sub>9</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"><a href="card_3.html#K5">pi</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> ,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>8</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"><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_1.html#T29" target="_self">Th29</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E6:49_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> 
 
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E1:49"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E7:49_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="lfuzzy_1.html#K4" target="_self">iter</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>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5">pi</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 </span>}</span> ,<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>
<a NAME="E8:49_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="lfuzzy_1.html#K4" target="_self">iter</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>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="lfuzzy_0.html#NR1" target="_self">&lt;&lt;=</a> <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K5" target="_self">TrCl</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>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E4:49_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/>
<b>hence </b><a NAME="E9:49_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lfuzzy_1.html#K4" target="_self">iter</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K5" target="_self">TrCl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lfuzzy_0.html#K7">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="lfuzzy_1.html#E2:49_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="lfuzzy_0.html#T3">LFUZZY_0:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:49"/>
 <a href="lfuzzy_1.html#K5" target="_self">TrCl</a> <font color="Maroon">c<sub>2</sub></font> <a href="fuzzy_1.html#NR2" target="_self">c=</a> 
 <b>by </b><i><a class="ref" href="fuzzy_1.html#D3">FUZZY_1:def 3</a></i>;<br/>


</div>
