<?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>, <font color="Maroon">c<sub>3</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>assume </b><a NAME="E1:25"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="lfuzzy_1.html#V3" target="_self">transitive</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="lfuzzy_1.html#V3" target="_self">transitive</a> )
 ;<br/>

<a NAME="E2:25"/><b>then </b><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="fuzzy_1.html#NR2" target="_self">c=</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_1.html#NR2" target="_self">c=</a>  )
 
<b>by </b><i><a class="ref" href="lfuzzy_1.html#D6" target="_self">Def6</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lfuzzy_1.html#D1" target="_self">LFUZZY_1:def 1</a><br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E3:25"/>
 <a href="lfuzzy_1.html#K1" target="_self">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fuzzy_1.html#NR2" target="_self">c=</a> 
 
<b>by </b><i><a class="ref" href="fuzzy_4.html#T15">FUZZY_4:15</a></i>;<br/>
<a NAME="E4:25"/><b>then </b><i><font color="Green">E19</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D3">FUZZY_1:def 3</a></i>;<br/>
<a NAME="E5:25"/>
(  <a href="lfuzzy_1.html#K1" target="_self">min</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="fuzzy_1.html#NR2" target="_self">c=</a>  &amp;  <a href="lfuzzy_1.html#K1" target="_self">min</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fuzzy_1.html#NR2" target="_self">c=</a>  )
 
<b>by </b><i><a class="ref" href="fuzzy_4.html#T16">FUZZY_4:16</a></i>;<br/>
<a NAME="E6:25"/><b>then </b>
( <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</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="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>4</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>3</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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>3</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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D4">FUZZY_1:def 4</a>, <a class="ref" href="fuzzy_1.html#D3">FUZZY_1:def 3</a></i>;<br/>
<a NAME="E7:25"/><b>then </b>
<span class="p1">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></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>4</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"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<a NAME="E8:25"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></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>4</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"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E4:25"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:25"/>
( <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <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="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>4</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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <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="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>3</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>4</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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>3</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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> )
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D4">FUZZY_1:def 4</a></i>;<br/>
<a NAME="E10:25"/><b>then </b>
( <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>3</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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> <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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>3</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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E2:25"><i><font color="Green">E18</font></i></a>, <a class="ref" href="lfuzzy_1.html#D1" target="_self">Def1</a>, <a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/>
<a NAME="E11:25"/><b>then </b>
( <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> <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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:25"/><b>then </b>
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></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>4</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"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_4.html#K3">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <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>4</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>3</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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<a NAME="E13:25"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <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>4</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>3</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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="lfuzzy_1.html#E8:25"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E14:25"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="fuzzy_4.html#K3">(#)</a> <span class="p2">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="lfuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="fuzzy_1.html#D4">FUZZY_1:def 4</a></i>;<br/>


</div>
