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

<b>let </b><font color="Maroon" title="c1">C2</font>, <font color="Maroon" title="c2">C3</font>, <font color="Maroon" title="c3">C1</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="fuzzy_3.html#NM1" title="FUZZY_3:NM.1">RMembership_Func</a> of <font color="Maroon" title="c1">C2</font>,<font color="Maroon" title="c2">C3</font>;<br/><b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>









<b>assume </b><a NAME="E1:32"/><i><font color="Green" title="E19">A1</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C1</font> &amp; <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C3</font> )
 ;<br/>

<a NAME="E2:32"/><i><font color="Green" title="E20">A2</font></i>: 
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p3">(<span class="default"><a href="fuzzy_3.html#K1" title="FUZZY_3:func.1">Zmf</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c1">C2</font></span>)</span>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_3.html#K1" title="FUZZY_3:func.1">Zmf</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c2">C3</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p2">(<span class="default"><a href="fuzzy_3.html#K1" title="FUZZY_3:func.1">Zmf</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c1">C2</font></span>)</span>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</font></span>)</span> is <a href="seq_4.html#V3" title="SEQ_4:attr.3">bounded</a>
 
<b>by </b><i><a class="ref" href="fuzzy_4.html#T1" target="_self" title="FUZZY_4:th.1">Th1</a></i>;<br/>
<a NAME="E2:32_1"/><b>then </b><i><font color="Green" title="E21">A3</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p2">(<span class="default"><a href="fuzzy_3.html#K1" title="FUZZY_3:func.1">Zmf</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c1">C2</font></span>)</span>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</font></span>)</span> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#D3" title="SEQ_4:def.3">SEQ_4:def 3</a></i>;<br/>
<a NAME="E3:32_1"/>
for <font color="Olive" title="b1">s</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <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="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p4">(<span class="default"><a href="fuzzy_3.html#K1" title="FUZZY_3:func.1">Zmf</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c1">C2</font></span>)</span>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">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_3.html#K1" title="FUZZY_3:func.1">Zmf</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c2">C3</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">s</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:32_1_1"/>
<font color="Maroon" title="c7">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:32_1_1"/><i><font color="Green" title="E22">A4</font></i>: 
( <font color="Maroon" title="c8">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p2">(<span class="default"><a href="fuzzy_3.html#K1" title="FUZZY_3:func.1">Zmf</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c1">C2</font></span>)</span>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p4">(<span class="default"><a href="fuzzy_3.html#K1" title="FUZZY_3:func.1">Zmf</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c1">C2</font></span>)</span>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r</font> )
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E2:32_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:32_1_1"/><i><font color="Green" title="E23">A5</font></i>: 
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p2">(<span class="default"><a href="fuzzy_3.html#K1" title="FUZZY_3:func.1">Zmf</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c1">C2</font></span>)</span>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</font></span>)</span> &amp; <font color="Maroon" title="c8">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> <span class="p2">(<span class="default"><a href="fuzzy_3.html#K1" title="FUZZY_3:func.1">Zmf</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c1">C2</font></span>)</span>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">y</font> )
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E3:32_1_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:32_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">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="c3">C1</font>,<font color="Maroon" title="c1">C2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c6">z</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">C2</font>,<font color="Maroon" title="c2">C3</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</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="c3">C1</font>,<font color="Maroon" title="c2">C3</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="txt" href="fuzzy_4.html#E1:32"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="fuzzy_4.html#E5:32_1_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E7:32_1_1"/><b>then </b><i><font color="Green" title="E24">A6</font></i>: 
( <span class="p1">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c1">C2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">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#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c2">C3</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></i>;<br/>
<a NAME="E8:32_1_1"/><i><font color="Green" title="E25">A7</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c6">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="ref" href="fuzzy_4.html#T3" target="_self" title="FUZZY_4:th.3">Th3</a></i>;<br/>
<font color="Maroon" title="c8">r</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_3.html#K1" title="FUZZY_3:func.1">Zmf</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c1">C2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">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="c4">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c6">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="fuzzy_4.html#E1:32"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="fuzzy_4.html#E5:32_1_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="fuzzy_4.html#D2" target="_self" title="FUZZY_4:def.2">Def2</a></i>
<br/>.= 
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c6">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="fuzzy_4.html#E7:32_1_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="fuzzy_3.html#D1" title="FUZZY_3:def.1">FUZZY_3:def 1</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="fuzzy_4.html#E8:32_1_1"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="xxreal_0.html#D8" title="XXREAL_0:def.8">XXREAL_0:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fuzzy_3.html#K1" title="FUZZY_3:func.1">Zmf</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c2">C3</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="fuzzy_4.html#E7:32_1_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="fuzzy_3.html#D1" title="FUZZY_3:def.1">FUZZY_3:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E10:32_1_1"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p4">(<span class="default"><a href="fuzzy_3.html#K1" title="FUZZY_3:func.1">Zmf</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c1">C2</font></span>)</span>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_3.html#K1" title="FUZZY_3:func.1">Zmf</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c2">C3</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E3:32_1_1"><i><font color="Green" title="E22">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:32_1"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p3">(<span class="default"><a href="fuzzy_3.html#K1" title="FUZZY_3:func.1">Zmf</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c1">C2</font></span>)</span>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_3.html#K1" title="FUZZY_3:func.1">Zmf</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c2">C3</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="real_2.html#T182" title="REAL_2:th.182">REAL_2:182</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:32"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p3">(<span class="default"><a href="fuzzy_3.html#K1" title="FUZZY_3:func.1">Zmf</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c1">C2</font></span>)</span>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_3.html#K1" title="FUZZY_3:func.1">Zmf</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c2">C3</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><div class="add">

<a NAME="E1:32_2"/><i><font color="Green" title="E21">A8</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p2">(<span class="default"><a href="fuzzy_3.html#K1" title="FUZZY_3:func.1">Zmf</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c1">C2</font></span>)</span>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="fuzzy_1.html#D1" title="FUZZY_1:def.1">FUZZY_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">A</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as  <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="integra1.html#D1" title="INTEGRA1:def.1">INTEGRA1:def 1</a></i>;<br/>
<a NAME="E3:32_2"/>
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c7">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c7">A</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></i>;<br/>
<a NAME="E4:32_2"/><b>then </b><i><font color="Green" title="E22">A9</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c7">A</font> &amp; 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c7">A</font> )
 
<b>by </b><i><a class="ref" href="integra1.html#T6" title="INTEGRA1:th.6">INTEGRA1:6</a></i>;<br/>
<a NAME="E5:32_2"/>
<font color="Maroon" title="c7">A</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 
<b>by </b><i><a class="ref" href="integra1.html#T3" title="INTEGRA1:th.3">INTEGRA1:3</a></i>;<br/>
<a NAME="E6:32_2"/><b>then </b><i><font color="Green" title="E23">A10</font></i>: 
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c7">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p3">(<span class="default"><a href="fuzzy_3.html#K1" title="FUZZY_3:func.1">Zmf</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c1">C2</font></span>)</span>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="fuzzy_4.html#E1:32_2"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="seq_4.html#T64" title="SEQ_4:th.64">SEQ_4:64</a></i>;<br/>
<a NAME="E7:32_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p2">(<span class="default"><a href="fuzzy_3.html#K1" title="FUZZY_3:func.1">Zmf</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c1">C2</font></span>)</span>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</font></span>)</span> is <a href="seq_4.html#V3" title="SEQ_4:attr.3">bounded</a>
 
<b>by </b><i><a class="ref" href="fuzzy_4.html#T1" target="_self" title="FUZZY_4:th.1">Th1</a></i>;<br/>
<a NAME="E8:32_2"/><b>then </b><i><font color="Green" title="E24">A11</font></i>: 
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p3">(<span class="default"><a href="fuzzy_3.html#K1" title="FUZZY_3:func.1">Zmf</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c1">C2</font></span>)</span>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p3">(<span class="default"><a href="fuzzy_3.html#K1" title="FUZZY_3:func.1">Zmf</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c1">C2</font></span>)</span>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_4.html#T24" title="SEQ_4:th.24">SEQ_4:24</a></i>;<br/>
<a NAME="E9:32_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</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="c3">C1</font>,<font color="Maroon" title="c2">C3</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="fuzzy_4.html#E1:32"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E10:32_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c2">C3</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></i>;<br/>
<b>hence </b><a NAME="E11:32_2"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p3">(<span class="default"><a href="fuzzy_3.html#K1" title="FUZZY_3:func.1">Zmf</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c1">C2</font></span>)</span>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="fuzzy_3.html#K1" title="FUZZY_3:func.1">Zmf</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c2">C3</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E4:32_2"><i><font color="Green" title="E22">A9</font></i></a>, <a class="txt" href="fuzzy_4.html#E6:32_2"><i><font color="Green" title="E23">A10</font></i></a>, <a class="txt" href="fuzzy_4.html#E8:32_2"><i><font color="Green" title="E24">A11</font></i></a>, <a class="ref" href="fuzzy_3.html#D1" title="FUZZY_3:def.1">FUZZY_3:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:32"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="fuzzy_4.html#K3" title="FUZZY_4:func.3">min</a> <span class="p3">(<span class="default"><a href="fuzzy_3.html#K1" title="FUZZY_3:func.1">Zmf</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c1">C2</font></span>)</span>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fuzzy_3.html#K1" title="FUZZY_3:func.1">Zmf</a> <font color="Maroon" title="c3">C1</font>,<font color="Maroon" title="c2">C3</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="fuzzy_4.html#E2:32"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


</div>
