<?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/><b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>











<b>assume </b><a NAME="E1:20"/><i><font color="Green" title="E12">A1</font></i>: 
( <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> &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C3</font> )
 ;<br/>

<a NAME="E2:20"/><i><font color="Green" title="E13">A2</font></i>: 
 <a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <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>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</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"><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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<a NAME="E1:20_1"/><i><font color="Green" title="E14">A3</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <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>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">C2</font> &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"><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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">C2</font> )
 
<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="E2:20_1"/>
for <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C2</font> holds <br/><span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <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>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">y</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#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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C2</font>;<br/>

<b>assume </b><a NAME="E1:20_1_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C2</font>
 ;<br/>

<a NAME="E2:20_1_1"/><i><font color="Green" title="E15">A4</font></i>: 
<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="c9">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>
 
<b>by </b><i><a class="txt" href="fuzzy_4.html#E1:20"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <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>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">y</font> = 
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</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="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="fuzzy_4.html#E1:20"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="fuzzy_4.html#D2" target="_self" title="FUZZY_4:def.2">Def2</a></i>
<br/>.= 
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="fuzzy_4.html#E2:20_1_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="fuzzy_1.html#D5" title="FUZZY_1:def.5">FUZZY_1:def 5</a></i>
<br/>.= 
 <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xxreal_0.html#T38" title="XXREAL_0:th.38">XXREAL_0:38</a></i>
<br/>.= 
 <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">y</font></span>)</span>
<b>by </b><i><a class="txt" href="fuzzy_4.html#E1:20"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="fuzzy_4.html#D2" target="_self" title="FUZZY_4:def.2">Def2</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"><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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">y</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">y</font></span>)</span>
<b>by </b><i><a class="txt" href="fuzzy_4.html#E1:20"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="fuzzy_4.html#D2" target="_self" title="FUZZY_4:def.2">Def2</a></i>
;<br/>
<b>hence </b><a NAME="E4:20_1_1"/>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <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>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">y</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#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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">y</font>
 <b>by </b><i><a class="ref" href="fuzzy_1.html#D5" title="FUZZY_1:def.5">FUZZY_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20_1"/>
 <a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <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>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</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"><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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E1:20_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c9">F</font> =  <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="c7">x</font>,<font color="Maroon" title="c8">z</font>;<br/>
<b>set </b><font color="Maroon" title="c10">G</font> =  <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="c7">x</font>,<font color="Maroon" title="c8">z</font>;<br/>
<a NAME="E3:20"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</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#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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> is <a href="seq_4.html#V3" title="SEQ_4:attr.3">bounded</a>
 
<b>by </b><i><a class="ref" href="fuzzy_4.html#T1" target="_self" title="FUZZY_4:th.1">Th1</a></i>;<br/>
<a NAME="E4:20"/><b>then </b><i><font color="Green" title="E14">A5</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</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#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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#D3" title="SEQ_4:def.3">SEQ_4:def 3</a></i>;<br/>
<a NAME="E5:20"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> is <a href="seq_4.html#V3" title="SEQ_4:attr.3">bounded</a>
 
<b>by </b><i><a class="ref" href="fuzzy_4.html#T1" target="_self" title="FUZZY_4:th.1">Th1</a></i>;<br/>
<a NAME="E6:20"/><b>then </b><i><font color="Green" title="E15">A6</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#D3" title="SEQ_4:def.3">SEQ_4:def 3</a></i>;<br/>
<a NAME="E7:20"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> is <a href="seq_4.html#V3" title="SEQ_4:attr.3">bounded</a>
 
<b>by </b><i><a class="ref" href="fuzzy_4.html#T1" target="_self" title="FUZZY_4:th.1">Th1</a></i>;<br/>
<a NAME="E8:20"/><b>then </b><i><font color="Green" title="E16">A7</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#D3" title="SEQ_4:def.3">SEQ_4:def 3</a></i>;<br/>
<a NAME="E9:20"/><i><font color="Green" title="E17">A8</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> holds <br/><span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:20_2"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>
 ;<br/>

<a NAME="E2:20_2"/><b>then </b>
<span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:20_2"/>
<span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E6:20"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:20"/><i><font color="Green" title="E18">A9</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> holds <br/><span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:20_3"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>
 ;<br/>

<a NAME="E2:20_3"/><b>then </b>
<span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:20_3"/>
<span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E8:20"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:20"/><i><font color="Green" title="E19">A10</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> holds <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#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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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_1.html#K2" title="FUZZY_1:func.2">max</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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:20_4"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:20_4"/><b>then </b>
<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#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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</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#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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:20_4"/>
<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#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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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_1.html#K2" title="FUZZY_1:func.2">max</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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E4:20"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:20"/><i><font color="Green" title="E20">A11</font></i>: 
 <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_1.html#K2" title="FUZZY_1:func.2">max</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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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="c7">x</font>,<font color="Maroon" title="c8">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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_5"><b>proof </b></a><div class="add">

<a NAME="E1:20_5"/>
for <font color="Olive" title="b1">s</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> holds <br/><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_1.html#K2" title="FUZZY_1:func.2">max</a> <span class="p4">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p4">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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="c7">x</font>,<font color="Maroon" title="c8">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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">s</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:20_5_1"/>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">s</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:20_5_1"/><i><font color="Green" title="E21">A12</font></i>: 
( <font color="Maroon" title="c12">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</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#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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> &amp; <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_1.html#K2" title="FUZZY_1:func.2">max</a> <span class="p4">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p4">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font> )
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E4:20"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:20_5_1"/><i><font color="Green" title="E22">A13</font></i>: 
( <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c12">r</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#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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">y</font> )
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E3:20_5_1"><i><font color="Green" title="E21">A12</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:20_5_1"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C2</font>
 
<b>by </b><i><a class="txt" href="fuzzy_4.html#E5:20_5_1"><i><font color="Green" title="E22">A13</font></i></a></i>;<br/>
<a NAME="E7:20_5_1"/><b>then </b>
( <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</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="E8:20_5_1"/><b>then </b>
( <span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> &amp; <span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="fuzzy_4.html#E9:20"><i><font color="Green" title="E17">A8</font></i></a>, <a class="txt" href="fuzzy_4.html#E10:20"><i><font color="Green" title="E18">A9</font></i></a></i>;<br/>
<a NAME="E9:20_5_1"/><b>then </b><i><font color="Green" title="E23">A14</font></i>: 
 <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#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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">y</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">y</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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="c7">x</font>,<font color="Maroon" title="c8">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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T26" title="XXREAL_0:th.26">XXREAL_0:26</a></i>;<br/>
<a NAME="E10:20_5_1"/>
<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_1.html#K2" title="FUZZY_1:func.2">max</a> <span class="p4">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p4">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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#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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">y</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="fuzzy_4.html#E3:20_5_1"><i><font color="Green" title="E21">A12</font></i></a>, <a class="txt" href="fuzzy_4.html#E5:20_5_1"><i><font color="Green" title="E22">A13</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="E11:20_5_1"/>
<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_1.html#K2" title="FUZZY_1:func.2">max</a> <span class="p4">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p4">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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="c7">x</font>,<font color="Maroon" title="c8">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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E9:20_5_1"><i><font color="Green" title="E23">A14</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:20_5"/>
 <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_1.html#K2" title="FUZZY_1:func.2">max</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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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="c7">x</font>,<font color="Maroon" title="c8">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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="real_2.html#T182" title="REAL_2:th.182">REAL_2:182</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:20"/>
 <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_1.html#K2" title="FUZZY_1:func.2">max</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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <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="c7">x</font>,<font color="Maroon" title="c8">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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_6"><b>proof </b></a><div class="add">

<a NAME="E1:20_6"/><i><font color="Green" title="E21">A15</font></i>: 
 <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> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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_1.html#K2" title="FUZZY_1:func.2">max</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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_6_1"><b>proof </b></a><div class="add">

<a NAME="E1:20_6_1"/><i><font color="Green" title="E22">A16</font></i>: 
for <font color="Olive" title="b1">s</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> &amp; <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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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#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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_6_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">s</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:20_6_1_1"/>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">s</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:20_6_1_1"/><i><font color="Green" title="E23">A17</font></i>: 
( <font color="Maroon" title="c12">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> &amp; <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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font> )
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E6:20"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:20_6_1_1"/><i><font color="Green" title="E24">A18</font></i>: 
( <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> &amp; <font color="Maroon" title="c12">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">y</font> )
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E3:20_6_1_1"><i><font color="Green" title="E23">A17</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:20_6_1_1"/>
<span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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#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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">y</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/>
<a NAME="E7:20_6_1_1"/><b>then </b>
<font color="Maroon" title="c12">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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#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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">y</font>
 
<b>by </b><i><a class="txt" href="fuzzy_4.html#E5:20_6_1_1"><i><font color="Green" title="E24">A18</font></i></a>, <a class="ref" href="fuzzy_1.html#D5" title="FUZZY_1:def.5">FUZZY_1:def 5</a></i>;<br/>
<a NAME="E8:20_6_1_1"/><b>then </b>
<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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#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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">y</font>
 
<b>by </b><i><a class="txt" href="fuzzy_4.html#E3:20_6_1_1"><i><font color="Green" title="E23">A17</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E9:20_6_1_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> &amp; <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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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#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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">y</font> )
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E5:20_6_1_1"><i><font color="Green" title="E24">A18</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:20_6_1"/>
for <font color="Olive" title="b1">s</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> holds <br/><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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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_1.html#K2" title="FUZZY_1:func.2">max</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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_6_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">s</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:20_6_1_2"/>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">s</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:20_6_1_2"/><i><font color="Green" title="E23">A19</font></i>: 
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> &amp; <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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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#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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">y</font> )
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E1:20_6_1"><i><font color="Green" title="E22">A16</font></i></a></i>;<br/>
<a NAME="E4:20_6_1_2"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C2</font>
 
<b>by </b><i><a class="txt" href="fuzzy_4.html#E3:20_6_1_2"><i><font color="Green" title="E23">A19</font></i></a></i>;<br/>
<a NAME="E5:20_6_1_2"/><b>then </b>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</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="E6:20_6_1_2"/><b>then </b>
<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#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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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_1.html#K2" title="FUZZY_1:func.2">max</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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="fuzzy_4.html#E11:20"><i><font color="Green" title="E19">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:20_6_1_2"/>
<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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_1.html#K2" title="FUZZY_1:func.2">max</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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E3:20_6_1_2"><i><font color="Green" title="E23">A19</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20_6_1"/>
 <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> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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_1.html#K2" title="FUZZY_1:func.2">max</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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="real_2.html#T182" title="REAL_2:th.182">REAL_2:182</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:20_6"/>
 <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> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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_1.html#K2" title="FUZZY_1:func.2">max</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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_6_2"><b>proof </b></a><div class="add">

<a NAME="E1:20_6_2"/><i><font color="Green" title="E22">A20</font></i>: 
for <font color="Olive" title="b1">s</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> &amp; <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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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#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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_6_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">s</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:20_6_2_1"/>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">s</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:20_6_2_1"/><i><font color="Green" title="E23">A21</font></i>: 
( <font color="Maroon" title="c12">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> &amp; <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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font> )
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E8:20"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:20_6_2_1"/><i><font color="Green" title="E24">A22</font></i>: 
( <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> &amp; <font color="Maroon" title="c12">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">y</font> )
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E3:20_6_2_1"><i><font color="Green" title="E23">A21</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:20_6_2_1"/>
<span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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#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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">y</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/>
<a NAME="E7:20_6_2_1"/><b>then </b>
<font color="Maroon" title="c12">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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#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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">y</font>
 
<b>by </b><i><a class="txt" href="fuzzy_4.html#E5:20_6_2_1"><i><font color="Green" title="E24">A22</font></i></a>, <a class="ref" href="fuzzy_1.html#D5" title="FUZZY_1:def.5">FUZZY_1:def 5</a></i>;<br/>
<a NAME="E8:20_6_2_1"/><b>then </b>
<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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#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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">y</font>
 
<b>by </b><i><a class="txt" href="fuzzy_4.html#E3:20_6_2_1"><i><font color="Green" title="E23">A21</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E9:20_6_2_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> &amp; <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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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#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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">y</font> )
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E5:20_6_2_1"><i><font color="Green" title="E24">A22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:20_6_2"/>
for <font color="Olive" title="b1">s</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> holds <br/><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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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_1.html#K2" title="FUZZY_1:func.2">max</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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_6_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">s</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:20_6_2_2"/>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">s</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:20_6_2_2"/><i><font color="Green" title="E23">A23</font></i>: 
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span> &amp; <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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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#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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">y</font> )
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E1:20_6_2"><i><font color="Green" title="E22">A20</font></i></a></i>;<br/>
<a NAME="E4:20_6_2_2"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C2</font>
 
<b>by </b><i><a class="txt" href="fuzzy_4.html#E3:20_6_2_2"><i><font color="Green" title="E23">A23</font></i></a></i>;<br/>
<a NAME="E5:20_6_2_2"/><b>then </b>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</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="E6:20_6_2_2"/><b>then </b>
<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#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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<span class="p2">(<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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_1.html#K2" title="FUZZY_1:func.2">max</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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="fuzzy_4.html#E11:20"><i><font color="Green" title="E19">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:20_6_2_2"/>
<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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_1.html#K2" title="FUZZY_1:func.2">max</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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E3:20_6_2_2"><i><font color="Green" title="E23">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20_6_2"/>
 <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> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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_1.html#K2" title="FUZZY_1:func.2">max</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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="real_2.html#T182" title="REAL_2:th.182">REAL_2:182</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20_6"/>
 <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_1.html#K2" title="FUZZY_1:func.2">max</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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span>,<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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <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="c7">x</font>,<font color="Maroon" title="c8">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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E1:20_6"><i><font color="Green" title="E21">A15</font></i></a>, <a class="ref" href="square_1.html#T50" title="SQUARE_1:th.50">SQUARE_1:50</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:20"/>
 <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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c7">x</font>,<font color="Maroon" title="c8">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="c7">x</font>,<font color="Maroon" title="c8">z</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E2:20"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="fuzzy_4.html#E12:20"><i><font color="Green" title="E20">A11</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


</div>
