<?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/>

<a NAME="E1:47"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K3" target="_self">1_minus</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K5" target="_self">UMF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E2:47"/>
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> 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/><span class="p1">(<span class="default"><a href="fuzzy_1.html#K3" target="_self">1_minus</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K5" target="_self">UMF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<span class="p1">(<span class="default"><a href="fuzzy_1.html#K3" target="_self">1_minus</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> = 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="fuzzy_1.html#D6" target="_self">Def6</a></i>
<br/>.= 
1 <a href="real_1.html#K5">-</a> 0
<b>by </b><i><a class="ref" href="fuzzy_1.html#T43" target="_self">Th43</a></i>
<br/>.= 
1

;<br/>
<b>hence </b><a NAME="E2:47_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> implies <span class="p1">(<span class="default"><a href="fuzzy_1.html#K3" target="_self">1_minus</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K5" target="_self">UMF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="fuzzy_1.html#T43" target="_self">Th43</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:47"/>
 <a href="fuzzy_1.html#K3" target="_self">1_minus</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fuzzy_1.html#K5" target="_self">UMF</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="fuzzy_1.html#E1:47"><i><font color="Green">E28</font></i></a>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>

<a NAME="E4:47"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K3" target="_self">1_minus</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K5" target="_self">UMF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E5:47"/>
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> 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/><span class="p1">(<span class="default"><a href="fuzzy_1.html#K3" target="_self">1_minus</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K5" target="_self">UMF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<span class="p1">(<span class="default"><a href="fuzzy_1.html#K3" target="_self">1_minus</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K5" target="_self">UMF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> = 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_1.html#K5" target="_self">UMF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="fuzzy_1.html#D6" target="_self">Def6</a></i>
<br/>.= 
1 <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="ref" href="fuzzy_1.html#T43" target="_self">Th43</a></i>
<br/>.= 
0

;<br/>
<b>hence </b><a NAME="E2:47_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> implies <span class="p1">(<span class="default"><a href="fuzzy_1.html#K3" target="_self">1_minus</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K5" target="_self">UMF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="fuzzy_1.html#T43" target="_self">Th43</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:47"/>
 <a href="fuzzy_1.html#K3" target="_self">1_minus</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K5" target="_self">UMF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fuzzy_1.html#K4" target="_self">EMF</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="fuzzy_1.html#E4:47"><i><font color="Green">E29</font></i></a>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


</div>
