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

<span class="kw">let </span><font color="Maroon" title="c1">C1</font>, <font color="Maroon" title="c2">C2</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="fuzzy_2.html#NM1" title="FUZZY_2:NM.1">RMembership_Func</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> holds   <a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font>)</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> (<span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <font color="Olive" title="b1">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <font color="Olive" title="b2">g</font></span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g</font> be   <a href="fuzzy_2.html#NM1" title="FUZZY_2:NM.1">RMembership_Func</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> (<span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <font color="Maroon" title="c4">g</font></span>)</span>)</span><br/>





<a NAME="E1:10"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> (<span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <font color="Maroon" title="c4">g</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">C2</font>,<font color="Maroon" title="c1">C1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E2:10"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
 for <font color="Olive" title="b1">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">C2</font>,<font color="Maroon" title="c1">C1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">C2</font>,<font color="Maroon" title="c1">C1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> (<span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <font color="Maroon" title="c4">g</font></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">C2</font>,<font color="Maroon" title="c1">C1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">C2</font>,<font color="Maroon" title="c1">C1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> implies <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> (<span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <font color="Maroon" title="c4">g</font></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">c</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:10_1"/>
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">C2</font>,<font color="Maroon" title="c1">C1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> (<span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <font color="Maroon" title="c4">g</font></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">c</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">y</font>, <font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:10_1"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C2</font>
 <span class="kw">and </span><br/><a NAME="E4:10_1"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font>
 <span class="kw">and </span><br/><a NAME="E5:10_1"/><span class="lab"><font color="Green" title="E10">A5</font></span>: 
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E6:10_1"/><span class="lab"><font color="Green" title="E11">A6</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_1"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:10_1"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="fuzzy_4.html#K1" title="FUZZY_4:func.1">.</a> (<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">x</font>) = 
<span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> <a href="fuzzy_4.html#K1" title="FUZZY_4:func.1">.</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c6">y</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10_1"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="ref" href="fuzzy_4.html#D1" target="_self" title="FUZZY_4:def.1">Def1</a></span>
<br/>.= 
 <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="fuzzy_4.html#K1" title="FUZZY_4:func.1">.</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c6">y</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="fuzzy_4.html#K1" title="FUZZY_4:func.1">.</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c6">y</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:10_1"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="ref" href="fuzzy_1.html#D4" title="FUZZY_1:def.4">FUZZY_1:def 4</a></span>
<br/>.= 
 <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="fuzzy_4.html#K1" title="FUZZY_4:func.1">.</a> (<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">x</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="fuzzy_4.html#K1" title="FUZZY_4:func.1">.</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c6">y</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10_1"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="ref" href="fuzzy_4.html#D1" target="_self" title="FUZZY_4:def.1">Def1</a></span>
<br/>.= 
 <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="fuzzy_4.html#K1" title="FUZZY_4:func.1">.</a> (<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">x</font>)</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="fuzzy_4.html#K1" title="FUZZY_4:func.1">.</a> (<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">x</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10_1"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="ref" href="fuzzy_4.html#D1" target="_self" title="FUZZY_4:def.1">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> (<span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <font color="Maroon" title="c4">g</font></span>)</span>)</span>)</span> <a href="fuzzy_4.html#K1" title="FUZZY_4:func.1">.</a> (<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">x</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10_1"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="ref" href="fuzzy_1.html#D4" title="FUZZY_1:def.4">FUZZY_1:def 4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:10_1"/>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> (<span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <font color="Maroon" title="c4">g</font></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10_1"><span class="lab"><font color="Green" title="E10">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:10"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <span class="p2">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">C2</font>,<font color="Maroon" title="c1">C1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:10"/>
 <a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font>)</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> (<span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="fuzzy_4.html#K2" title="FUZZY_4:func.2">converse</a> <font color="Maroon" title="c4">g</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="partfun1.html#T5" title="PARTFUN1:th.5">PARTFUN1:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
