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

<span class="kw">let </span><font color="Maroon" title="c1">C1</font>, <font color="Maroon" title="c2">C2</font>, <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="fuzzy_2.html#NM1" title="FUZZY_2:NM.1">RMembership_Func</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font><br/>  for <font color="Olive" title="b3">h</font> being   <a href="fuzzy_2.html#NM1" title="FUZZY_2:NM.1">RMembership_Func</a> of <font color="Maroon" title="c2">C2</font>,<font color="Maroon" title="c3">C3</font><br/>  for <font color="Olive" title="b4">x</font>, <font color="Olive" title="b5">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> &amp; <font color="Olive" title="b5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C3</font> holds <br/> <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font>)</span>)</span>,<font color="Olive" title="b3">h</font>,<font color="Olive" title="b4">x</font>,<font color="Olive" title="b5">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">h</font>,<font color="Olive" title="b4">x</font>,<font color="Olive" title="b5">z</font>)</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> (<font color="Olive" title="b2">g</font>,<font color="Olive" title="b3">h</font>,<font color="Olive" title="b4">x</font>,<font color="Olive" title="b5">z</font>)</span>)</span></span>)</span></span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font> be   <a href="fuzzy_2.html#NM1" title="FUZZY_2:NM.1">RMembership_Func</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">h</font> being   <a href="fuzzy_2.html#NM1" title="FUZZY_2:NM.1">RMembership_Func</a> of <font color="Maroon" title="c2">C2</font>,<font color="Maroon" title="c3">C3</font><br/>  for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> &amp; <font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C3</font> holds <br/> <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="Olive" title="b1">h</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="Olive" title="b1">h</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">z</font>)</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="Olive" title="b1">h</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">z</font>)</span>)</span></span>)</span></span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c6">h</font> be   <a href="fuzzy_2.html#NM1" title="FUZZY_2:NM.1">RMembership_Func</a> of <font color="Maroon" title="c2">C2</font>,<font color="Maroon" title="c3">C3</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> &amp; <font color="Olive" title="b2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C3</font> holds <br/> <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="Olive" title="b1">x</font>,<font color="Olive" title="b2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="Olive" title="b1">x</font>,<font color="Olive" title="b2">z</font>)</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="Olive" title="b1">x</font>,<font color="Olive" title="b2">z</font>)</span>)</span></span>)</span></span>)</span>)</span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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> implies  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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>) )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:20"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font>
 <span class="kw">and </span><br/><a NAME="E2:20"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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>)</span><br/>

<a NAME="E3:20"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
 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#K1" title="SEQ_1:func.1">.</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#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C2</font> implies <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#K1" title="SEQ_1:func.1">.</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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:20_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="fuzzy_4.html#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#K1" title="SEQ_1:func.1">.</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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">y</font></span><br/>

<a NAME="E2:20_1"/><span class="lab"><font color="Green" title="E15">A4</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<span class="p1">(<span class="default"><a href="fuzzy_4.html#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#K1" title="SEQ_1:func.1">.</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#K1" title="SEQ_1:func.1">.</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#K1" title="SEQ_1:func.1">.</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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="fuzzy_4.html#D2" target="_self" title="FUZZY_4:def.2">Def2</a></span>
<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#K1" title="SEQ_1:func.1">.</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#K1" title="SEQ_1:func.1">.</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#K1" title="SEQ_1:func.1">.</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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="ref" href="fuzzy_1.html#D4" title="FUZZY_1:def.4">FUZZY_1:def 4</a></span>
<br/>.= 
 <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> (<span class="p1">(<span class="default"><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#K1" title="SEQ_1:func.1">.</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#K1" title="SEQ_1:func.1">.</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#K1" title="SEQ_1:func.1">.</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#K1" title="SEQ_1:func.1">.</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="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T38" title="XXREAL_0:th.38">XXREAL_0:38</a></span>
<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#K1" title="SEQ_1:func.1">.</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#K1" title="SEQ_1:func.1">.</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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">y</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="fuzzy_4.html#D2" target="_self" title="FUZZY_4:def.2">Def2</a></span>
<br/>.= 
 <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fuzzy_4.html#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#K1" title="SEQ_1:func.1">.</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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">y</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="fuzzy_4.html#D2" target="_self" title="FUZZY_4:def.2">Def2</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:20_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#K1" title="SEQ_1:func.1">.</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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fuzzy_1.html#D4" title="FUZZY_1:def.4">FUZZY_1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">set </span><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/>
<span class="kw">set </span><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="E4:20"/><span class="lab"><font color="Green" title="E15">A5</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> (<span class="p2">(<span class="default"><a href="fuzzy_4.html#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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E5:20"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fuzzy_4.html#T1" target="_self" title="FUZZY_4:th.1">Th1</a></span>;<br/>
<a NAME="E6:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A6</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D11" title="XXREAL_2:def.11">XXREAL_2:def 11</a></span>;<br/>
<a NAME="E7:20"/><span class="lab"><font color="Green" title="E17">A7</font></span>: 
 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#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> (<span class="p2">(<span class="default"><a href="fuzzy_4.html#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#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> (<span class="p2">(<span class="default"><a href="fuzzy_4.html#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> implies <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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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> )</span><br/>

<span class="kw">assume </span><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#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> (<span class="p2">(<span class="default"><a href="fuzzy_4.html#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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="fuzzy_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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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></span><br/>

<a NAME="E2:20_2"/><span class="kw">then </span>
<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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:20_2"/>
<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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="ref" href="seq_4.html#D1" title="SEQ_4:def.1">SEQ_4:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:20"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fuzzy_4.html#T1" target="_self" title="FUZZY_4:th.1">Th1</a></span>;<br/>
<a NAME="E9:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A8</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D11" title="XXREAL_2:def.11">XXREAL_2:def 11</a></span>;<br/>
<a NAME="E10:20"/><span class="lab"><font color="Green" title="E19">A9</font></span>: 
 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#K1" title="RELSET_1:func.1">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#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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> implies <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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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> )</span><br/>

<span class="kw">assume </span><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#K1" title="RELSET_1:func.1">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="fuzzy_4.html#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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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></span><br/>

<a NAME="E2:20_3"/><span class="kw">then </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> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">hence </span><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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="seq_4.html#D1" title="SEQ_4:def.1">SEQ_4:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E11:20"/><span class="lab"><font color="Green" title="E20">A10</font></span>: 
 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  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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#K1" title="RELSET_1:func.1">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">s</font> implies  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#K1" title="RELSET_1:func.1">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">y</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:20_4"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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#K1" title="RELSET_1:func.1">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">y</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:20_4"/><span class="lab"><font color="Green" title="E21">A11</font></span>: 
<font color="Maroon" title="c12">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 <span class="kw">and </span><br/><a NAME="E4:20_4"/><span class="lab"><font color="Green" title="E22">A12</font></span>: 
<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:20"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="seq_4.html#D1" title="SEQ_4:def.1">SEQ_4:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:20_4"/><span class="lab"><font color="Green" title="E23">A13</font></span>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 <span class="kw">and </span><br/><a NAME="E7:20_4"/><span class="lab"><font color="Green" title="E24">A14</font></span>: 
<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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_4"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E8:20_4"/>
<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#K1" title="SEQ_1:func.1">.</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#K1" title="SEQ_1:func.1">.</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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">y</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<a NAME="E9:20_4"/><span class="kw">then </span>
<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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20_4"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="txt" href="#E7:20_4"><span class="lab"><font color="Green" title="E24">A14</font></span></a>, <a class="ref" href="fuzzy_1.html#D4" title="FUZZY_1:def.4">FUZZY_1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:20_4"/>
 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#K1" title="RELSET_1:func.1">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_4"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E6:20_4"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:20"/>
 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  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">s</font> implies <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:20_5"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:20_5"/><span class="lab"><font color="Green" title="E21">A15</font></span>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 <span class="kw">and </span><br/><a NAME="E4:20_5"/><span class="lab"><font color="Green" title="E22">A16</font></span>: 
<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:20"><span class="lab"><font color="Green" title="E20">A10</font></span></a></span>;<br/>
<a NAME="E5:20_5"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_5"><span class="lab"><font color="Green" title="E21">A15</font></span></a></span>;<br/>
<a NAME="E6:20_5"/><span class="kw">then </span>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> (<span class="p2">(<span class="default"><a href="fuzzy_4.html#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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E7:20_5"/><span class="kw">then </span>
<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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20"><span class="lab"><font color="Green" title="E17">A7</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:20_5"/>
<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_5"><span class="lab"><font color="Green" title="E22">A16</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A17</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T57" title="XREAL_1:th.57">XREAL_1:57</a></span>;<br/>
<a NAME="E14:20"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fuzzy_4.html#T1" target="_self" title="FUZZY_4:th.1">Th1</a></span>;<br/>
<a NAME="E15:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A18</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D11" title="XXREAL_2:def.11">XXREAL_2:def 11</a></span>;<br/>
<a NAME="E16:20"/><span class="lab"><font color="Green" title="E23">A19</font></span>: 
 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  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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#K1" title="RELSET_1:func.1">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">s</font> implies  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#K1" title="RELSET_1:func.1">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">y</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:20_6"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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#K1" title="RELSET_1:func.1">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">y</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:20_6"/><span class="lab"><font color="Green" title="E24">A20</font></span>: 
<font color="Maroon" title="c12">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 <span class="kw">and </span><br/><a NAME="E4:20_6"/><span class="lab"><font color="Green" title="E25">A21</font></span>: 
<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:20"><span class="lab"><font color="Green" title="E22">A18</font></span></a>, <a class="ref" href="seq_4.html#D1" title="SEQ_4:def.1">SEQ_4:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:20_6"/><span class="lab"><font color="Green" title="E26">A22</font></span>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 <span class="kw">and </span><br/><a NAME="E7:20_6"/><span class="lab"><font color="Green" title="E27">A23</font></span>: 
<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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_6"><span class="lab"><font color="Green" title="E24">A20</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E8:20_6"/>
<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#K1" title="SEQ_1:func.1">.</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#K1" title="SEQ_1:func.1">.</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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">y</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<a NAME="E9:20_6"/><span class="kw">then </span>
<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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20_6"><span class="lab"><font color="Green" title="E26">A22</font></span></a>, <a class="txt" href="#E7:20_6"><span class="lab"><font color="Green" title="E27">A23</font></span></a>, <a class="ref" href="fuzzy_1.html#D4" title="FUZZY_1:def.4">FUZZY_1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:20_6"/>
 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#K1" title="RELSET_1:func.1">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_6"><span class="lab"><font color="Green" title="E25">A21</font></span></a>, <a class="txt" href="#E6:20_6"><span class="lab"><font color="Green" title="E26">A22</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E17:20"/>
 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  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">s</font> implies <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:20_7"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:20_7"/><span class="lab"><font color="Green" title="E24">A24</font></span>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 <span class="kw">and </span><br/><a NAME="E4:20_7"/><span class="lab"><font color="Green" title="E25">A25</font></span>: 
<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:20"><span class="lab"><font color="Green" title="E23">A19</font></span></a></span>;<br/>
<a NAME="E5:20_7"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_7"><span class="lab"><font color="Green" title="E24">A24</font></span></a></span>;<br/>
<a NAME="E6:20_7"/><span class="kw">then </span>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> (<span class="p2">(<span class="default"><a href="fuzzy_4.html#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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E7:20_7"/><span class="kw">then </span>
<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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:20"><span class="lab"><font color="Green" title="E17">A7</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:20_7"/>
<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_7"><span class="lab"><font color="Green" title="E25">A25</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E18:20"/><span class="kw">then </span>
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T57" title="XREAL_1:th.57">XREAL_1:57</a></span>;<br/>
<a NAME="E19:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A26</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:20"><span class="lab"><font color="Green" title="E21">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T28" title="XXREAL_0:th.28">XXREAL_0:28</a></span>;<br/>
<a NAME="E20:20"/><span class="lab"><font color="Green" title="E25">A27</font></span>: 
 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#K1" title="RELSET_1:func.1">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#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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> implies <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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:20_8"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="fuzzy_4.html#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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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></span><br/>

<a NAME="E2:20_8"/><span class="kw">then </span>
<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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:20_8"/>
<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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:20"><span class="lab"><font color="Green" title="E22">A18</font></span></a>, <a class="ref" href="seq_4.html#D1" title="SEQ_4:def.1">SEQ_4:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E21:20"/>
 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  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">s</font> implies <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:20_9"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:20_9"/><span class="lab"><font color="Green" title="E26">A28</font></span>: 
<font color="Maroon" title="c12">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 <span class="kw">and </span><br/><a NAME="E4:20_9"/><span class="lab"><font color="Green" title="E27">A29</font></span>: 
<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="ref" href="seq_4.html#D1" title="SEQ_4:def.1">SEQ_4:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:20_9"/><span class="lab"><font color="Green" title="E28">A30</font></span>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_1.html#K2" title="FUZZY_1:func.2">max</a> (<span class="p2">(<span class="default"><a href="fuzzy_4.html#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>
 <span class="kw">and </span><br/><a NAME="E7:20_9"/><span class="lab"><font color="Green" title="E29">A31</font></span>: 
<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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_9"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E8:20_9"/><span class="lab"><font color="Green" title="E30">A32</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20_9"><span class="lab"><font color="Green" title="E28">A30</font></span></a></span>;<br/>
<a NAME="E9:20_9"/><span class="kw">then </span>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E10:20_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A33</font></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> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20"><span class="lab"><font color="Green" title="E19">A9</font></span></a></span>;<br/>
<a NAME="E11:20_9"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20_9"><span class="lab"><font color="Green" title="E30">A32</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E12:20_9"/><span class="kw">then </span>
<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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:20"><span class="lab"><font color="Green" title="E25">A27</font></span></a></span>;<br/>
<a NAME="E13:20_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A34</font></span>: 
 <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#K1" title="SEQ_1:func.1">.</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#K1" title="SEQ_1:func.1">.</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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20_9"><span class="lab"><font color="Green" title="E31">A33</font></span></a>, <a class="ref" href="xxreal_0.html#T26" title="XXREAL_0:th.26">XXREAL_0:26</a></span>;<br/>
<a NAME="E14:20_9"/>
<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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#K1" title="SEQ_1:func.1">.</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#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">y</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_9"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E6:20_9"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E7:20_9"><span class="lab"><font color="Green" title="E29">A31</font></span></a>, <a class="ref" href="fuzzy_1.html#D4" title="FUZZY_1:def.4">FUZZY_1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:20_9"/>
<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:20_9"><span class="lab"><font color="Green" title="E32">A34</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E22:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A35</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T57" title="XREAL_1:th.57">XREAL_1:57</a></span>;<br/>
<a NAME="E23:20"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E24:20"/><span class="kw">then </span>
 <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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="partfun1.html#T5" title="PARTFUN1:th.5">PARTFUN1:5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:20"/>
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:20"><span class="lab"><font color="Green" title="E26">A35</font></span></a>, <a class="txt" href="#E19:20"><span class="lab"><font color="Green" title="E24">A26</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
