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

<b>let </b><font color="Maroon">C1</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">C2</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">f</font> be   <a href="fuzzy_3.html#NM1">RMembership_Func</a> of <font color="Maroon">C1</font>,<font color="Maroon">C2</font>;<br/>





<a NAME="E1:6"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">c</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">C1</font>,<font color="Maroon">C2</font></span><a href="zfmisc_1.html#K2">:]</a></span>  st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">C1</font>,<font color="Maroon">C2</font></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><span class="p1">(<span class="default"><a href="fuzzy_4.html#K1" target="_self">converse</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K1" target="_self">converse</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</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">C1</font>,<font color="Maroon">C2</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>

<b>assume </b><a NAME="E1:6_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">C1</font>,<font color="Maroon">C2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E4:6_1"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">C2</font>,<font color="Maroon">C1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E5:6_1"/>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K1" target="_self">converse</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K1" target="_self">converse</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K1" target="_self">converse</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:6_1"/>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K1" target="_self">converse</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K1" target="_self">converse</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:6"/>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K1" target="_self">converse</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K1" target="_self">converse</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">C1</font>,<font color="Maroon">C2</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">C1</font>,<font color="Maroon">C2</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D1">FUZZY_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:6"/>
 <a href="fuzzy_4.html#K1" target="_self">converse</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K1" target="_self">converse</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


</div>
