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

<b>let </b><font color="Maroon" title="c1">C1</font>, <font color="Maroon" title="c2">C2</font>, <font color="Maroon" title="c3">C3</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> ;<br/><b>let </b><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font> be   <a href="fuzzy_3.html#NM1" title="FUZZY_3:NM.1">RMembership_Func</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>;<br/><b>let </b><font color="Maroon" title="c6">h</font> be   <a href="fuzzy_3.html#NM1" title="FUZZY_3:NM.1">RMembership_Func</a> of <font color="Maroon" title="c2">C2</font>,<font color="Maroon" title="c3">C3</font>;<br/>









<a NAME="E1:21"/><i><font color="Green" title="E12">A1</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</font></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="c1">C1</font>,<font color="Maroon" title="c3">C3</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</font></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="c1">C1</font>,<font color="Maroon" title="c3">C3</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">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"><font color="Maroon" title="c4">f</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</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="c1">C1</font>,<font color="Maroon" title="c3">C3</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</font></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="c1">C1</font>,<font color="Maroon" title="c3">C3</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D1" title="FUZZY_1:def.1">FUZZY_1:def 1</a></i>;<br/>
<a NAME="E2:21"/>
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="c1">C1</font>,<font color="Maroon" title="c3">C3</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="c1">C1</font>,<font color="Maroon" title="c3">C3</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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"><font color="Maroon" title="c4">f</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">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="c1">C1</font>,<font color="Maroon" title="c3">C3</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>

<b>consider </b><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:21_1"/><i><font color="Green" title="E13">A2</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> &amp; <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C3</font> &amp; <font color="Maroon" title="c7">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="c8">x</font>,<font color="Maroon" title="c9">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">c</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</font></span>)</span> <a href="fuzzy_4.html#K1" title="FUZZY_4:func.1">.</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">z</font>
<b>by </b><i><a class="txt" href="fuzzy_4.html#E2:21_1"><i><font color="Green" title="E13">A2</font></i></a></i>
<br/>.= 
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p3">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fuzzy_4.html#E2:21_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="fuzzy_4.html#D3" target="_self" title="FUZZY_4:def.3">Def3</a></i>
<br/>.= 
 <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">z</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">z</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fuzzy_4.html#E2:21_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="fuzzy_4.html#E18"><i><font color="Green" title="E11">Lm2</font></i></a></i>
<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"><font color="Maroon" title="c4">f</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</font></span>)</span> <a href="fuzzy_4.html#K1" title="FUZZY_4:func.1">.</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">z</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">z</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fuzzy_4.html#E2:21_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="fuzzy_4.html#D3" target="_self" title="FUZZY_4:def.3">Def3</a></i>
<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"><font color="Maroon" title="c4">f</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</font></span>)</span> <a href="fuzzy_4.html#K1" title="FUZZY_4:func.1">.</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">z</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</font></span>)</span> <a href="fuzzy_4.html#K1" title="FUZZY_4:func.1">.</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">z</font></span>)</span>
<b>by </b><i><a class="txt" href="fuzzy_4.html#E2:21_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="fuzzy_4.html#D3" target="_self" title="FUZZY_4:def.3">Def3</a></i>
<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"><font color="Maroon" title="c4">f</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">c</font>
<b>by </b><i><a class="txt" href="fuzzy_4.html#E2:21_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="fuzzy_1.html#D5" title="FUZZY_1:def.5">FUZZY_1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E4:21_1"/>
( <font color="Maroon" title="c7">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="c1">C1</font>,<font color="Maroon" title="c3">C3</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">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"><font color="Maroon" title="c4">f</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">c</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:21"/>
<span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</font> <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"><font color="Maroon" title="c4">f</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="fuzzy_4.html#K4" title="FUZZY_4:func.4">(#)</a> <font color="Maroon" title="c6">h</font></span>)</span>
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E1:21"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>;<br/>


</div>
