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<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>, <font color="Maroon">c<sub>4</sub></font> be    <a href="fuzzy_1.html#M1" target="_self">Membership_Func</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:40_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="fuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>hence </b><a NAME="E2:40_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_1.html#NR2" target="_self">c=</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="fuzzy_1.html#NR2" target="_self">c=</a>  )
 <b>by </b><i><a class="ref" href="fuzzy_1.html#T18" target="_self">Th18</a></i>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="fuzzy_1.html#M1" target="_self">Membership_Func</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E3:40_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_1.html#NR2" target="_self">c=</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="fuzzy_1.html#NR2" target="_self">c=</a>  )
 ;<br/><b>thus </b><a NAME="E4:40_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="fuzzy_1.html#NR2" target="_self">c=</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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="fuzzy_1.html#D3" target="_self">FUZZY_1:def 3</a><br/>

<a NAME="E1:40_1_1"/>
( <font color="Maroon">c<sub>5</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> <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>5</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> <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="txt" href="fuzzy_1.html#E3:40_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="fuzzy_1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E2:40_1_1"/><b>then </b>
 <a href="square_1.html#K1">min</a> <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>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</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="square_1.html#T39">SQUARE_1:39</a></i>;<br/>
<b>hence </b><a NAME="E3:40_1_1"/>
<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#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="fuzzy_1.html#E1:40_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="fuzzy_1.html#D4" target="_self">Def4</a></i>;<br/>


</div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><b>that </b><br/><a NAME="E2:40"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_1.html#NR2" target="_self">c=</a> 
 <b>and </b><br/><a NAME="E3:40"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="fuzzy_1.html#NR2" target="_self">c=</a> 
 <b>and </b><br/><a NAME="E4:40"/><i><font color="Green">E26</font></i>: 
for <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 <font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_1.html#NR2" target="_self">c=</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="fuzzy_1.html#NR2" target="_self">c=</a>  holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="fuzzy_1.html#NR2" target="_self">c=</a> 
 ;<br/>

<a NAME="E5:40"/><i><font color="Green">E27</font></i>: 
 <a href="fuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="fuzzy_1.html#NR2" target="_self">c=</a> 
 
<b>by </b><i><a class="txt" href="fuzzy_1.html#E2:40"><i><font color="Green">E24</font></i></a>, <a class="txt" href="fuzzy_1.html#E3:40"><i><font color="Green">E25</font></i></a>, <a class="ref" href="fuzzy_1.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E6:40"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="fuzzy_1.html#NR2" target="_self">c=</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="fuzzy_1.html#NR2" target="_self">c=</a>  )
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E7:40"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="fuzzy_1.html#NR2" target="_self">c=</a> 
 
<b>by </b><i><a class="txt" href="fuzzy_1.html#E4:40"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:40"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="fuzzy_1.html#K1" target="_self">min</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="fuzzy_1.html#E5:40"><i><font color="Green">E27</font></i></a>, <a class="txt" href="fuzzy_1.html#E6"><i><font color="Green">Lemma6</font></i></a></i>;<br/>


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