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

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1_1_1"/>
( <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> or  not <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_1"><b>suppose </b></a><a NAME="E1:83_1_1_1_1"/><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">f</font> = <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="binarith.html#K11">*&gt;</a></span>;<br/><a NAME="E2:83_1_1_1_1"/><i><font color="Green">E47</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/><b>thus </b><a NAME="E3:83_1_1_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:83_1_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>hence </b><a NAME="E2:83_1_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T18" target="_self">Th18</a>, <a class="ref" href="rcomp_3.html#T19" target="_self">Th19</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E4:83_1_1_1_1"/>
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T19" target="_self">Th19</a>, <a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>;<br/><b>thus </b><a NAME="E5:83_1_1_1_1"/>
( ( for <font color="Olive">n</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/>( ( <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> implies  not <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> is <a href="xboole_0.html#V1">empty</a> ) &amp; ( <font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> implies (  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font></span>)</span> ) ) &amp; ( <font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> implies  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span></span>)</span> ) ) ) &amp; ( <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> implies <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="binarith.html#K11">*&gt;</a></span> ) &amp; ( not <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> implies (  ex <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">p</font></span><a href="rcomp_2.html#K1">.[</a></span> ) &amp;  ex <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">p</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> ) &amp; ( for <font color="Olive">n</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> holds <br/> ex <font color="Olive">p</font>, <font color="Olive">q</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">p</font>,<font color="Olive">q</font></span><a href="rcomp_1.html#K2">.[</a></span> ) ) ) ) )
 <b>by </b><i><a class="ref" href="rcomp_3.html#T17" target="_self">Th17</a>, <a class="ref" href="rcomp_3.html#T18" target="_self">Th18</a>, <a class="ref" href="rcomp_3.html#T5" target="_self">Th5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2"><b>suppose </b></a><a NAME="E1:83_1_1_1_2"/><i><font color="Green">E48</font></i>: 
not <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><div class="add"><a NAME="E2:83_1_1_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T17" target="_self">Th17</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E3:83_1_1_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T17" target="_self">Th17</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>set </b><font color="Maroon">L</font> =  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font>;<br/><a NAME="E4:83_1_1_1_2"/><i><font color="Green">E51</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T17" target="_self">Th17</a>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/><a NAME="E5:83_1_1_1_2"/><i><font color="Green">E52</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon">T</font> =  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> as  non <a href="struct_0.html#V3">empty</a> <a href="connsp_1.html#V1">connected</a> <a href="pre_topc.html#NM1">TopSpace</a> <b>by </b><i><a class="ref" href="rcomp_3.html#T17" target="_self">Th17</a>, <a class="ref" href="treal_1.html#T23">TREAL_1:23</a></i>;<br/><a NAME="E7:83_1_1_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">F</font> <a href="compts_1.html#R1">is_a_cover_of</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T12" target="_self">Th12</a>, <a class="txt" href="rcomp_3.html#E4:83_1_1_1_2"><i><font color="Green">E51</font></i></a></i>;<br/><a NAME="E8:83_1_1_1_2"/>
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> is <a href="compts_1.html#V2">compact</a>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T17" target="_self">Th17</a>, <a class="ref" href="heine.html#T11">HEINE:11</a></i>;<br/><a NAME="E9:83_1_1_1_2"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> is <a href="compts_1.html#V6">compact</a>
 <b>by </b><i><a class="ref" href="compts_1.html#T10">COMPTS_1:10</a></i>;<br/><b>then consider </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span><b> such that </b><br/><a NAME="E11:83_1_1_1_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E12:83_1_1_1_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">G</font> <a href="compts_1.html#R1">is_a_cover_of</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>
 <b>and </b><br/><a NAME="E13:83_1_1_1_2"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">G</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T21" target="_self">Th21</a>, <a class="ref" href="rcomp_3.html#T14" target="_self">Th14</a>, <a class="ref" href="compts_1.html#D7">COMPTS_1:def 7</a></i>;<br/><a NAME="E14:83_1_1_1_2"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">G</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E4:83_1_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="rcomp_3.html#T23" target="_self">Th23</a></i>;<br/><a NAME="E15:83_1_1_1_2"/><b>then </b><i><font color="Green">E98</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>;<br/><div><i><font color="Green">E99</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>;<br/><a NAME="E1:83_1_1_1_2_1"/>
<font color="Maroon">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E2:83_1_1_1_2_1"/>
( <font color="Maroon">A</font> is <a href="seq_4.html#V1">bounded_above</a> &amp; <font color="Maroon">A</font> is <a href="seq_4.html#V2">bounded_below</a> )
 <b>by </b><i><a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="rcomp_1.html#T3">RCOMP_1:3</a>, <a class="ref" href="rcomp_1.html#T4">RCOMP_1:4</a></i>;<br/><b>hence </b><a NAME="E3:83_1_1_1_2_1"/>
<font color="Maroon">A</font> is  <a href="seq_4.html#V3">bounded</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, <a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/></div><b>end;</b></div><b>set </b><font color="Maroon">ZAW</font> = <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> ;<br/><b>set </b><font color="Maroon">G1</font> = <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> ;<br/><a NAME="E17:83_1_1_1_2"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><a NAME="E18:83_1_1_1_2"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T24" target="_self">Th24</a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><div><i><font color="Green">E102</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_1_1_2_2"/><i><font color="Green">E103</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> 
 ;<br/><a NAME="E2:83_1_1_1_2_2"/><b>then </b><i><font color="Green">E104</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:83_1_1_1_2_2"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>consider </b><font color="Maroon">g</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:83_1_1_1_2_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E7:83_1_1_1_2_2"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T33" target="_self">Th33</a>, <a class="ref" href="rcomp_3.html#T26" target="_self">Th26</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E8:83_1_1_1_2_2"/>
(  <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">g</font> &amp;  <a href="xboole_0.html#K1">{}</a>  is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  )
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><a NAME="E9:83_1_1_1_2_2"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_3.html#T32" target="_self">Th32</a>, <a class="ref" href="rcomp_3.html#T35" target="_self">Th35</a>, <a class="ref" href="rcomp_3.html#T34" target="_self">Th34</a></i>;<br/><b>hence </b><a NAME="E10:83_1_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="ref" href="rcomp_3.html#T31" target="_self">Th31</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><a NAME="E20:83_1_1_1_2"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T22" target="_self">Th22</a>, <a class="ref" href="rcomp_3.html#T28" target="_self">Th28</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E21:83_1_1_1_2"/><i><font color="Green">E109</font></i>: 
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  holds <br/><font color="Olive">X</font> is  <a href="jct_misc.html#V1">connected</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:83_1_1_1_2_3"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as  <a href="connsp_1.html#V2">connected</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> <b>by </b><i><a class="ref" href="rcomp_3.html#T16" target="_self">Th16</a>, <a class="ref" href="rcomp_3.html#T31" target="_self">Th31</a>, <a class="ref" href="rcomp_3.html#T4" target="_self">Th4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">X</font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E4:83_1_1_1_2_3"/>
<font color="Maroon">Y</font> is <a href="jct_misc.html#V1">connected</a>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E2:83_1_1_1_2"><i><font color="Green">E49</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:83_1_1_1_2_3"/>
<font color="Maroon">X</font> is  <a href="jct_misc.html#V1">connected</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E22:83_1_1_1_2"/>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K1">REAL</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:83_1_1_1_2_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:83_1_1_1_2_4"/><b>then </b>
<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E3:83_1_1_1_2_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K1">REAL</a> 
 ;<br/>


</div><b>end;</b></div><a NAME="E23:83_1_1_1_2"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="rcomp_3.html#T31" target="_self">Th31</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>set </b><font color="Maroon">ALL</font> = <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">C</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  ) </span>}</span> ;<br/><b>set </b><font color="Maroon">R</font> =  <a href="wellord2.html#K1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">C</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  ) </span>}</span> ;<br/><a NAME="E24:83_1_1_1_2"/><i><font color="Green">E111</font></i>: 
 <a href="wellord2.html#K1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">C</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  ) </span>}</span>  <a href="orders_1.html#R2">partially_orders</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">C</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:83_1_1_1_2_5"/>
(  <a href="wellord2.html#K1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">C</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  ) </span>}</span>  <a href="relat_2.html#R1">is_reflexive_in</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">C</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  ) </span>}</span>  &amp;  <a href="wellord2.html#K1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">C</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  ) </span>}</span>  <a href="relat_2.html#R8">is_transitive_in</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">C</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  ) </span>}</span>  &amp;  <a href="wellord2.html#K1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">C</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  ) </span>}</span>  <a href="relat_2.html#R4">is_antisymmetric_in</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">C</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="wellord2.html#T29">WELLORD2:29</a>, <a class="ref" href="wellord2.html#T30">WELLORD2:30</a>, <a class="ref" href="wellord2.html#T31">WELLORD2:31</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D7">ORDERS_1:def 7</a><br/>


</div><b>end;</b></div><a NAME="E25:83_1_1_1_2"/><i><font color="Green">E112</font></i>: 
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p2">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">C</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  ) </span>}</span> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">C</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  ) </span>}</span> 
 <b>by </b><i><a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/><a NAME="E26:83_1_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">C</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  ) </span>}</span>  <a href="orders_1.html#R5">has_lower_Zorn_property_wrt</a>  <a href="wellord2.html#K1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">C</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_3.html#T25" target="_self">Th25</a>, <a class="ref" href="rcomp_3.html#T24" target="_self">Th24</a>, <a class="ref" href="rcomp_3.html#T6" target="_self">Th6</a></i>;<br/><b>then consider </b><font color="Maroon">M</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E28:83_1_1_1_2"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">M</font> <a href="orders_1.html#R7">is_minimal_in</a>  <a href="wellord2.html#K1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">C</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_3.html#T34" target="_self">Th34</a>, <a class="ref" href="rcomp_3.html#T35" target="_self">Th35</a>, <a class="ref" href="orders_1.html#T174">ORDERS_1:174</a></i>;<br/><a NAME="E29:83_1_1_1_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p2">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">C</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  ) </span>}</span> </span>)</span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T36" target="_self">Th36</a>, <a class="ref" href="orders_1.html#D12">ORDERS_1:def 12</a></i>;<br/><b>then consider </b><font color="Maroon">C</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span><b> such that </b><br/><a NAME="E31:83_1_1_1_2"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E32:83_1_1_1_2"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">C</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E33:83_1_1_1_2"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_3.html#T35" target="_self">Th35</a></i>;<br/><a NAME="E34:83_1_1_1_2"/><i><font color="Green">E118</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T39" target="_self">Th39</a>, <a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>;<br/><b>consider </b><font color="Maroon">R1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E36:83_1_1_1_2"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font>
 <b>and </b><br/><a NAME="E37:83_1_1_1_2"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">R1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="rcomp_3.html#T41" target="_self">Th41</a>, <a class="ref" href="rcomp_3.html#T19" target="_self">Th19</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">R2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E39:83_1_1_1_2"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font>
 <b>and </b><br/><a NAME="E40:83_1_1_1_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">R2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="rcomp_3.html#T41" target="_self">Th41</a>, <a class="ref" href="rcomp_3.html#T20" target="_self">Th20</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E41:83_1_1_1_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">R1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_3.html#T40" target="_self">Th40</a>, </i>;<br/><a NAME="E42:83_1_1_1_2"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">R1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T31" target="_self">Th31</a></i>;<br/><a NAME="E43:83_1_1_1_2"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">R2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_3.html#T40" target="_self">Th40</a>, <a class="txt" href="rcomp_3.html#E11:83_1_1_1_2"><i><font color="Green">E91</font></i></a></i>;<br/><a NAME="E44:83_1_1_1_2"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">R2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T31" target="_self">Th31</a></i>;<br/><b>reconsider </b><font color="Maroon">R1</font> = <font color="Maroon">R1</font>, <font color="Maroon">R2</font> = <font color="Maroon">R2</font> as  <a href="pre_topc.html#V3">open</a> <a href="connsp_1.html#V2">connected</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> <b>by </b><i><a class="ref" href="rcomp_3.html#T31" target="_self">Th31</a>, <a class="ref" href="rcomp_3.html#T43" target="_self">Th43</a>, , <a class="ref" href="rcomp_3.html#T14" target="_self">Th14</a>, <a class="ref" href="rcomp_3.html#T16" target="_self">Th16</a>, <a class="ref" href="rcomp_3.html#T4" target="_self">Th4</a>, <a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">X</font> = <font color="Maroon">C</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="rcomp_3.html#T29" target="_self">Th29</a>, <a class="ref" href="rcomp_3.html#T40" target="_self">Th40</a>, <a class="ref" href="rcomp_3.html#T33" target="_self">Th33</a>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><a NAME="E47:83_1_1_1_2"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T40" target="_self">Th40</a>, <a class="ref" href="rcomp_3.html#T31" target="_self">Th31</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E48:83_1_1_1_2"/><i><font color="Green">E161</font></i>: 
for <font color="Olive">Z</font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">Z</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="connsp_1.html#V2">connected</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Z</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:83_1_1_1_2_6"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:83_1_1_1_2_6"/><b>then </b>
( not <font color="Maroon">Z</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">Z</font> is <a href="jct_misc.html#V1">connected</a> )
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T40" target="_self">Th40</a>, <a class="ref" href="rcomp_3.html#T32" target="_self">Th32</a>, <a class="ref" href="rcomp_3.html#T30" target="_self">Th30</a></i>;<br/>
<b>hence </b><a NAME="E3:83_1_1_1_2_6"/>
<font color="Maroon">Z</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="connsp_1.html#V2">connected</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E13:83_1_1_1_2"><i><font color="Green">E96</font></i></a>, <a class="txt" href="rcomp_3.html#E15:83_1_1_1_2"><i><font color="Green">E98</font></i></a>, <a class="txt" href="rcomp_3.html#E2:83_1_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="rcomp_3.html#T14" target="_self">Th14</a>, <a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon">LM</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">b</font></span><a href="rcomp_2.html#K1">.[</a></span></span>)</span> where <font color="Olive">b</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">b</font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  </span>}</span> ;<br/><b>set </b><font color="Maroon">LM1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pscomp_1.html#NK2" target="_self">sup</a> <font color="Olive">E</font></span>)</span> where <font color="Olive">E</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : <font color="Olive">E</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  </span>}</span> ;<br/><a NAME="E49:83_1_1_1_2"/><i><font color="Green">E163</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pscomp_1.html#NK2" target="_self">sup</a> <font color="Olive">E</font></span>)</span> where <font color="Olive">E</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : <font color="Olive">E</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>(<a class="ref" href="rcomp_3.html#T29" target="_self">Th29</a>);<br/></i><a NAME="E50:83_1_1_1_2"/><i><font color="Green">E164</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">b</font></span><a href="rcomp_2.html#K1">.[</a></span></span>)</span> where <font color="Olive">b</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">b</font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pscomp_1.html#NK2" target="_self">sup</a> <font color="Olive">E</font></span>)</span> where <font color="Olive">E</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : <font color="Olive">E</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:83_1_1_1_2_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">b</font></span><a href="rcomp_2.html#K1">.[</a></span></span>)</span> where <font color="Olive">b</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">b</font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  </span>}</span> 
 ;<br/>

<a NAME="E2:83_1_1_1_2_7"/><b>then </b>
 ex <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">b</font></span><a href="rcomp_2.html#K1">.[</a></span> &amp; <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">b</font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  )
 
;<br/>
<b>hence </b><a NAME="E3:83_1_1_1_2_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pscomp_1.html#NK2" target="_self">sup</a> <font color="Olive">E</font></span>)</span> where <font color="Olive">E</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : <font color="Olive">E</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  </span>}</span> 
 ;<br/>


</div><b>end;</b></div><div><i><font color="Green">E165</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_8"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1_1_1_2_8_1"/>
(  ex <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">R1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">a</font></span><a href="rcomp_2.html#K1">.[</a></span> ) or  ex <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">R1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> ) or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">R1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K2">.[</a></span> ) )
 </span><b>by </b><i><a class="ref" href="rcomp_3.html#T17" target="_self">Th17</a>, , <a class="ref" href="rcomp_3.html#T18" target="_self">Th18</a>, , <a class="txt" href="rcomp_3.html#E3:83_1_1_1_2"><i><font color="Green">E50</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_8_1_1"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_8_1_1"/>
 ex <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">R1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">a</font></span><a href="rcomp_2.html#K1">.[</a></span> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">a</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:83_1_1_1_2_8_1_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E4:83_1_1_1_2_8_1_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E5:83_1_1_1_2_8_1_1"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">R1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">a</font></span><a href="rcomp_2.html#K1">.[</a></span>
 ;<br/><a NAME="E6:83_1_1_1_2_8_1_1"/>
<font color="Maroon">a</font> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E7:83_1_1_1_2_8_1_1"/><b>then </b>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">a</font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">b</font></span><a href="rcomp_2.html#K1">.[</a></span></span>)</span> where <font color="Olive">b</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">b</font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_3.html#T40" target="_self">Th40</a>, , <a class="txt" href="rcomp_3.html#E18:83_1_1_1_2"><i><font color="Green">E101</font></i></a></i>;<br/><b>hence </b><a NAME="E8:83_1_1_1_2_8_1_1"/>
not <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">b</font></span><a href="rcomp_2.html#K1">.[</a></span></span>)</span> where <font color="Olive">b</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">b</font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_8_1_2"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_8_1_2"/>
 ex <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">R1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:83_1_1_1_2_8_1_2"/>
not <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">b</font></span><a href="rcomp_2.html#K1">.[</a></span></span>)</span> where <font color="Olive">b</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">b</font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_8_1_3"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_8_1_3"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">R1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K2">.[</a></span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:83_1_1_1_2_8_1_3"/>
not <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">b</font></span><a href="rcomp_2.html#K1">.[</a></span></span>)</span> where <font color="Olive">b</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">b</font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E52:83_1_1_1_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">b</font></span><a href="rcomp_2.html#K1">.[</a></span></span>)</span> where <font color="Olive">b</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">b</font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:83_1_1_1_2_9"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">b</font></span><a href="rcomp_2.html#K1">.[</a></span></span>)</span> where <font color="Olive">b</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">b</font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  </span>}</span> 
 ;<br/>

<a NAME="E2:83_1_1_1_2_9"/><b>then </b>
 ex <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">b</font></span><a href="rcomp_2.html#K1">.[</a></span> &amp; <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">b</font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  )
 
;<br/>
<b>hence </b><a NAME="E3:83_1_1_1_2_9"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">LM</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">b</font></span><a href="rcomp_2.html#K1">.[</a></span></span>)</span> where <font color="Olive">b</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">b</font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="txt" href="rcomp_3.html#E16:83_1_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="rcomp_3.html#E15:83_1_1_1_2"><i><font color="Green">E98</font></i></a>, <a class="txt" href="rcomp_3.html#E17:83_1_1_1_2"><i><font color="Green">E100</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon">kL</font> =  <a href="pre_circ.html#K1">max</a> <font color="Maroon">LM</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>set </b><font color="Maroon">LEWY</font> = <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">kL</font></span><a href="rcomp_2.html#K1">.[</a></span>;<br/><a NAME="E55:83_1_1_1_2"/>
<font color="Maroon">kL</font> <a href="hidden.html#R2">in</a> <font color="Maroon">LM</font>
 <b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">b</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E57:83_1_1_1_2"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">kL</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">b</font></span><a href="rcomp_2.html#K1">.[</a></span>
 <b>and </b><br/><a NAME="E58:83_1_1_1_2"/><i><font color="Green">E190</font></i>: 
<span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">b</font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> 
 ;<br/><a NAME="E59:83_1_1_1_2"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E19:83_1_1_1_2"><i><font color="Green">E102</font></i></a>, <a class="ref" href="rcomp_3.html#T30" target="_self">Th30</a>, <a class="ref" href="rcomp_2.html#T9">RCOMP_2:9</a></i>;<br/><a NAME="E60:83_1_1_1_2"/><b>then </b><i><font color="Green">E192</font></i>: 
<font color="Maroon">kL</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E18:83_1_1_1_2"><i><font color="Green">E101</font></i></a>, </i>;<br/><a NAME="E61:83_1_1_1_2"/><i><font color="Green">E193</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">b</font></span><a href="rcomp_2.html#K1">.[</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/><a NAME="E62:83_1_1_1_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">kL</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E18:83_1_1_1_2"><i><font color="Green">E101</font></i></a>, , </i>;<br/><a NAME="E63:83_1_1_1_2"/><b>then </b><i><font color="Green">E194</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">kL</font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/><a NAME="E64:83_1_1_1_2"/><i><font color="Green">E195</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">kL</font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">kL</font>
 <b>by </b><i>, <a class="txt" href="rcomp_3.html#E18:83_1_1_1_2"><i><font color="Green">E101</font></i></a>, </i>;<br/><b>reconsider </b><font color="Maroon">LEWY</font> = <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">kL</font></span><a href="rcomp_2.html#K1">.[</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> <b>by </b><i><a class="txt" href="rcomp_3.html#E19:83_1_1_1_2"><i><font color="Green">E102</font></i></a>, , <a class="txt" href="rcomp_3.html#E18:83_1_1_1_2"><i><font color="Green">E101</font></i></a>, , </i>;<br/><b>set </b><font color="Maroon">RM</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">c</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span></span>)</span> where <font color="Olive">c</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">c</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  </span>}</span> ;<br/><b>set </b><font color="Maroon">RM1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pscomp_1.html#NK3" target="_self">inf</a> <font color="Olive">E</font></span>)</span> where <font color="Olive">E</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : <font color="Olive">E</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  </span>}</span> ;<br/><a NAME="E66:83_1_1_1_2"/><i><font color="Green">E198</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pscomp_1.html#NK3" target="_self">inf</a> <font color="Olive">E</font></span>)</span> where <font color="Olive">E</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : <font color="Olive">E</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>(<a class="ref" href="rcomp_3.html#T29" target="_self">Th29</a>);<br/></i><a NAME="E67:83_1_1_1_2"/><i><font color="Green">E199</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">c</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span></span>)</span> where <font color="Olive">c</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">c</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pscomp_1.html#NK3" target="_self">inf</a> <font color="Olive">E</font></span>)</span> where <font color="Olive">E</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : <font color="Olive">E</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:83_1_1_1_2_10"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">c</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span></span>)</span> where <font color="Olive">c</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">c</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  </span>}</span> 
 ;<br/>

<a NAME="E2:83_1_1_1_2_10"/><b>then </b>
 ex <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> &amp; <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  )
 
;<br/>
<b>hence </b><a NAME="E3:83_1_1_1_2_10"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pscomp_1.html#NK3" target="_self">inf</a> <font color="Olive">E</font></span>)</span> where <font color="Olive">E</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : <font color="Olive">E</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  </span>}</span> 
 ;<br/>


</div><b>end;</b></div><div><i><font color="Green">E200</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_11"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_11_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1_1_1_2_11_1"/>
(  ex <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">R2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">a</font></span><a href="rcomp_2.html#K1">.[</a></span> ) or  ex <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">R2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> ) or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">R2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K2">.[</a></span> ) )
 </span><b>by </b><i><a class="ref" href="rcomp_3.html#T17" target="_self">Th17</a>, <a class="ref" href="rcomp_3.html#T42" target="_self">Th42</a>, <a class="ref" href="rcomp_3.html#T18" target="_self">Th18</a>, <a class="txt" href="rcomp_3.html#E12:83_1_1_1_2"><i><font color="Green">E95</font></i></a>, <a class="txt" href="rcomp_3.html#E3:83_1_1_1_2"><i><font color="Green">E50</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_11_1_1"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_11_1_1"/>
 ex <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">R2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">a</font></span><a href="rcomp_2.html#K1">.[</a></span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:83_1_1_1_2_11_1_1"/>
not <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">c</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span></span>)</span> where <font color="Olive">c</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">c</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T42" target="_self">Th42</a>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_11_1_2"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_11_1_2"/>
 ex <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">R2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">a</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:83_1_1_1_2_11_1_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E4:83_1_1_1_2_11_1_2"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E5:83_1_1_1_2_11_1_2"/><i><font color="Green">E201</font></i>: 
<font color="Maroon">R2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span>
 ;<br/><a NAME="E6:83_1_1_1_2_11_1_2"/>
<font color="Maroon">a</font> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E7:83_1_1_1_2_11_1_2"/><b>then </b>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">c</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span></span>)</span> where <font color="Olive">c</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">c</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_3.html#T40" target="_self">Th40</a>, <a class="txt" href="rcomp_3.html#E11:83_1_1_1_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="rcomp_3.html#E24:83_1_1_1_2"><i><font color="Green">E111</font></i></a></i>;<br/><b>hence </b><a NAME="E8:83_1_1_1_2_11_1_2"/>
not <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">c</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span></span>)</span> where <font color="Olive">c</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">c</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_11_1_3"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_11_1_3"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">R2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K2">.[</a></span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:83_1_1_1_2_11_1_3"/>
not <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">c</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span></span>)</span> where <font color="Olive">c</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">c</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T42" target="_self">Th42</a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E69:83_1_1_1_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">c</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span></span>)</span> where <font color="Olive">c</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">c</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:83_1_1_1_2_12"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">c</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span></span>)</span> where <font color="Olive">c</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">c</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  </span>}</span> 
 ;<br/>

<a NAME="E2:83_1_1_1_2_12"/><b>then </b>
 ex <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> &amp; <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">b</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  )
 
;<br/>
<b>hence </b><a NAME="E3:83_1_1_1_2_12"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">RM</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">c</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span></span>)</span> where <font color="Olive">c</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">c</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="txt" href="rcomp_3.html#E21:83_1_1_1_2"><i><font color="Green">E109</font></i></a>, <a class="txt" href="rcomp_3.html#E20:83_1_1_1_2"><i><font color="Green">E108</font></i></a>, <a class="txt" href="rcomp_3.html#E23:83_1_1_1_2"><i><font color="Green">E110</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon">pP</font> =  <a href="seq_4.html#K3">min</a> <font color="Maroon">RM</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>set </b><font color="Maroon">PRAWY</font> = <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">pP</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span>;<br/><a NAME="E72:83_1_1_1_2"/>
<font color="Maroon">pP</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RM</font>
 <b>by </b><i><a class="ref" href="pre_circ.html#D4">PRE_CIRC:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">b</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E74:83_1_1_1_2"/><i><font color="Green">E204</font></i>: 
<font color="Maroon">pP</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span>
 <b>and </b><br/><a NAME="E75:83_1_1_1_2"/><i><font color="Green">E205</font></i>: 
<span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> 
 ;<br/><a NAME="E76:83_1_1_1_2"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E25:83_1_1_1_2"><i><font color="Green">E112</font></i></a>, <a class="ref" href="rcomp_3.html#T30" target="_self">Th30</a>, <a class="ref" href="rcomp_2.html#T10">RCOMP_2:10</a></i>;<br/><a NAME="E77:83_1_1_1_2"/><b>then </b><i><font color="Green">E207</font></i>: 
<font color="Maroon">pP</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E24:83_1_1_1_2"><i><font color="Green">E111</font></i></a>, </i>;<br/><a NAME="E78:83_1_1_1_2"/><i><font color="Green">E208</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E28:83_1_1_1_2"><i><font color="Green">E113</font></i></a>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/><a NAME="E79:83_1_1_1_2"/>
<font color="Maroon">pP</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E24:83_1_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="txt" href="rcomp_3.html#E28:83_1_1_1_2"><i><font color="Green">E113</font></i></a>, </i>;<br/><a NAME="E80:83_1_1_1_2"/><b>then </b><i><font color="Green">E209</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">pP</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/><a NAME="E81:83_1_1_1_2"/><i><font color="Green">E210</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">pP</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">pP</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E24:83_1_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="txt" href="rcomp_3.html#E28:83_1_1_1_2"><i><font color="Green">E113</font></i></a>, </i>;<br/><b>reconsider </b><font color="Maroon">PRAWY</font> = <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">pP</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> <b>by </b><i><a class="txt" href="rcomp_3.html#E25:83_1_1_1_2"><i><font color="Green">E112</font></i></a>, <a class="txt" href="rcomp_3.html#E31:83_1_1_1_2"><i><font color="Green">E115</font></i></a>, <a class="txt" href="rcomp_3.html#E24:83_1_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="txt" href="rcomp_3.html#E28:83_1_1_1_2"><i><font color="Green">E113</font></i></a>, </i>;<br/><a NAME="E83:83_1_1_1_2"/><i><font color="Green">E211</font></i>: 
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>  st <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  holds <br/><font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">LEWY</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:83_1_1_1_2_13"/><i><font color="Green">E212</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E2:83_1_1_1_2_13"/><i><font color="Green">E213</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> 
 ;<br/>

<b>assume </b><a NAME="E3:83_1_1_1_2_13"/><i><font color="Green">E214</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">LEWY</font>
 ;<br/>

<a NAME="E4:83_1_1_1_2_13"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E37:83_1_1_1_2"><i><font color="Green">E120</font></i></a>, <a class="ref" href="rcomp_3.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E5:83_1_1_1_2_13"/><i><font color="Green">E216</font></i>: 
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E37:83_1_1_1_2"><i><font color="Green">E120</font></i></a>, <a class="ref" href="rcomp_3.html#T31" target="_self">Th31</a>, <a class="ref" href="rcomp_3.html#T14" target="_self">Th14</a>, <a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>
<a NAME="E6:83_1_1_1_2_13"/><i><font color="Green">E217</font></i>: 
<font color="Maroon">A</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E37:83_1_1_1_2"><i><font color="Green">E120</font></i></a>, <a class="ref" href="rcomp_3.html#T31" target="_self">Th31</a>, <a class="ref" href="rcomp_3.html#T16" target="_self">Th16</a>, <a class="ref" href="rcomp_3.html#T4" target="_self">Th4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_13_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_1_1_2_13_1"/><i><font color="Green">E218</font></i>: 
(  ex <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> ) or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K2">.[</a></span> ) )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_13_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1_1_1_2_13_1_1"/>
(  ex <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> ) or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K2">.[</a></span> ) )
 </span><b>by </b><i><a class="txt" href="rcomp_3.html#E43:83_1_1_1_2"><i><font color="Green">E125</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_13_1_1_1"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_13_1_1_1"/>
 ex <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:83_1_1_1_2_13_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rcomp_3.html#E36:83_1_1_1_2"><i><font color="Green">E119</font></i></a>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_13_1_1_2"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_13_1_1_2"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K2">.[</a></span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:83_1_1_1_2_13_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="rcomp_3.html#E36:83_1_1_1_2"><i><font color="Green">E119</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">ak</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E9:83_1_1_1_2_13"/><i><font color="Green">E220</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ak</font> &amp; <font color="Maroon">ak</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">ak</font></span><a href="rcomp_2.html#K1">.[</a></span> )
 <b>by </b><i><a class="txt" href="rcomp_3.html#E36:83_1_1_1_2"><i><font color="Green">E119</font></i></a>, <a class="txt" href="rcomp_3.html#E40:83_1_1_1_2"><i><font color="Green">E122</font></i></a>, <a class="txt" href="rcomp_3.html#E41:83_1_1_1_2"><i><font color="Green">E123</font></i></a>, <a class="txt" href="rcomp_3.html#E42:83_1_1_1_2"><i><font color="Green">E124</font></i></a>, <a class="ref" href="rcomp_3.html#T17" target="_self">Th17</a>, <a class="ref" href="rcomp_3.html#T18" target="_self">Th18</a>, <a class="txt" href="rcomp_3.html#E3:83_1_1_1_2"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E10:83_1_1_1_2_13"/><i><font color="Green">E221</font></i>: 
<font color="Maroon">ak</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E11:83_1_1_1_2_13"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">LEWY</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_13_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:83_1_1_1_2_13_2"/><i><font color="Green">E222</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:83_1_1_1_2_13_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E4:83_1_1_1_2_13_2"/><i><font color="Green">E223</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E43:83_1_1_1_2"><i><font color="Green">E125</font></i></a>, <a class="ref" href="rcomp_3.html#T44" target="_self">Th44</a>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/>
<a NAME="E5:83_1_1_1_2_13_2"/><i><font color="Green">E224</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ak</font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E43:83_1_1_1_2"><i><font color="Green">E125</font></i></a>, <a class="ref" href="rcomp_3.html#T44" target="_self">Th44</a>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/>
<a NAME="E6:83_1_1_1_2_13_2"/>
 <a href="pscomp_1.html#NK2" target="_self">sup</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ak</font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E43:83_1_1_1_2"><i><font color="Green">E125</font></i></a>, </i>;<br/>
<a NAME="E7:83_1_1_1_2_13_2"/><b>then </b>
<font color="Maroon">ak</font> <a href="hidden.html#R2">in</a> <font color="Maroon">LM</font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E37:83_1_1_1_2"><i><font color="Green">E120</font></i></a>, <a class="txt" href="rcomp_3.html#E43:83_1_1_1_2"><i><font color="Green">E125</font></i></a>, <a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a></i>;<br/>
<a NAME="E8:83_1_1_1_2_13_2"/><b>then </b>
<font color="Maroon">ak</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kL</font>
 
<b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<a NAME="E9:83_1_1_1_2_13_2"/><b>then </b>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">kL</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E10:83_1_1_1_2_13_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">LEWY</font>
 <b>by </b><i>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:83_1_1_1_2_13"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E37:83_1_1_1_2"><i><font color="Green">E120</font></i></a>, <a class="txt" href="rcomp_3.html#E39:83_1_1_1_2"><i><font color="Green">E121</font></i></a>, <a class="ref" href="rcomp_3.html#T28" target="_self">Th28</a>, <a class="txt" href="rcomp_3.html#E19:83_1_1_1_2"><i><font color="Green">E102</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E13:83_1_1_1_2_13"/>
contradiction
 <b>by </b><i><a class="txt" href="rcomp_3.html#E37:83_1_1_1_2"><i><font color="Green">E120</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E84:83_1_1_1_2"/><i><font color="Green">E225</font></i>: 
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>  st <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  holds <br/><font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">PRAWY</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_14"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:83_1_1_1_2_14"/><i><font color="Green">E226</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E2:83_1_1_1_2_14"/><i><font color="Green">E227</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> 
 ;<br/>

<b>assume </b><a NAME="E3:83_1_1_1_2_14"/><i><font color="Green">E228</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">PRAWY</font>
 ;<br/>

<a NAME="E4:83_1_1_1_2_14"/><i><font color="Green">E229</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E39:83_1_1_1_2"><i><font color="Green">E121</font></i></a>, <a class="ref" href="rcomp_3.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E5:83_1_1_1_2_14"/><i><font color="Green">E230</font></i>: 
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E39:83_1_1_1_2"><i><font color="Green">E121</font></i></a>, <a class="ref" href="rcomp_3.html#T31" target="_self">Th31</a>, <a class="ref" href="rcomp_3.html#T14" target="_self">Th14</a>, <a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>
<a NAME="E6:83_1_1_1_2_14"/><i><font color="Green">E231</font></i>: 
<font color="Maroon">A</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E39:83_1_1_1_2"><i><font color="Green">E121</font></i></a>, <a class="ref" href="rcomp_3.html#T31" target="_self">Th31</a>, <a class="ref" href="rcomp_3.html#T16" target="_self">Th16</a>, <a class="ref" href="rcomp_3.html#T4" target="_self">Th4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_14_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_1_1_2_14_1"/><i><font color="Green">E232</font></i>: 
(  ex <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">a</font></span><a href="rcomp_2.html#K1">.[</a></span> ) or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K2">.[</a></span> ) )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_14_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1_1_1_2_14_1_1"/>
(  ex <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">a</font></span><a href="rcomp_2.html#K1">.[</a></span> ) or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K2">.[</a></span> ) )
 </span><b>by </b><i><a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_14_1_1_1"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_14_1_1_1"/>
 ex <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">a</font></span><a href="rcomp_2.html#K1">.[</a></span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:83_1_1_1_2_14_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rcomp_3.html#E37:83_1_1_1_2"><i><font color="Green">E120</font></i></a>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_14_1_1_2"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_14_1_1_2"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K2">.[</a></span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:83_1_1_1_2_14_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="rcomp_3.html#E37:83_1_1_1_2"><i><font color="Green">E120</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">ak</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E9:83_1_1_1_2_14"/><i><font color="Green">E233</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ak</font> &amp; <font color="Maroon">ak</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">ak</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> )
 <b>by </b><i><a class="txt" href="rcomp_3.html#E37:83_1_1_1_2"><i><font color="Green">E120</font></i></a>, <a class="txt" href="rcomp_3.html#E41:83_1_1_1_2"><i><font color="Green">E123</font></i></a>, <a class="txt" href="rcomp_3.html#E42:83_1_1_1_2"><i><font color="Green">E124</font></i></a>, <a class="txt" href="rcomp_3.html#E43:83_1_1_1_2"><i><font color="Green">E125</font></i></a>, <a class="ref" href="rcomp_3.html#T17" target="_self">Th17</a>, <a class="ref" href="rcomp_3.html#T18" target="_self">Th18</a>, <a class="txt" href="rcomp_3.html#E3:83_1_1_1_2"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E10:83_1_1_1_2_14"/><i><font color="Green">E234</font></i>: 
<font color="Maroon">ak</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E11:83_1_1_1_2_14"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PRAWY</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_14_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:83_1_1_1_2_14_2"/><i><font color="Green">E235</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:83_1_1_1_2_14_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E4:83_1_1_1_2_14_2"/><i><font color="Green">E236</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a>, , <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/>
<a NAME="E5:83_1_1_1_2_14_2"/><i><font color="Green">E237</font></i>: 
<font color="Maroon">ak</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a>, , <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/>
<a NAME="E6:83_1_1_1_2_14_2"/>
 <a href="pscomp_1.html#NK3" target="_self">inf</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ak</font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a>, </i>;<br/>
<a NAME="E7:83_1_1_1_2_14_2"/><b>then </b>
<font color="Maroon">ak</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RM</font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E39:83_1_1_1_2"><i><font color="Green">E121</font></i></a>, <a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a>, <a class="ref" href="rcomp_3.html#T44" target="_self">Th44</a></i>;<br/>
<a NAME="E8:83_1_1_1_2_14_2"/><b>then </b>
<font color="Maroon">pP</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ak</font>
 
<b>by </b><i><a class="ref" href="pre_circ.html#D4">PRE_CIRC:def 4</a></i>;<br/>
<a NAME="E9:83_1_1_1_2_14_2"/><b>then </b>
<font color="Maroon">pP</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E10:83_1_1_1_2_14_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PRAWY</font>
 <b>by </b><i>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:83_1_1_1_2_14"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E39:83_1_1_1_2"><i><font color="Green">E121</font></i></a>, <a class="txt" href="rcomp_3.html#E40:83_1_1_1_2"><i><font color="Green">E122</font></i></a>, <a class="ref" href="rcomp_3.html#T28" target="_self">Th28</a>, <a class="txt" href="rcomp_3.html#E25:83_1_1_1_2"><i><font color="Green">E112</font></i></a>, <a class="txt" href="rcomp_3.html#E29:83_1_1_1_2"><i><font color="Green">E114</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:83_1_1_1_2_14"/>
contradiction
 <b>by </b><i><a class="txt" href="rcomp_3.html#E39:83_1_1_1_2"><i><font color="Green">E121</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E85:83_1_1_1_2"/><i><font color="Green">E238</font></i>: 
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">LEWY</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">PRAWY</font> holds <br/> ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K2">.[</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_15"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:83_1_1_1_2_15"/><i><font color="Green">E239</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> 
 <b>and </b><br/><a NAME="E2:83_1_1_1_2_15"/><i><font color="Green">E240</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">LEWY</font>
 <b>and </b><br/><a NAME="E3:83_1_1_1_2_15"/><i><font color="Green">E241</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">PRAWY</font>
 ;<br/>

<a NAME="E4:83_1_1_1_2_15"/><i><font color="Green">E242</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E39:83_1_1_1_2"><i><font color="Green">E121</font></i></a>, <a class="ref" href="rcomp_3.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E5:83_1_1_1_2_15"/><i><font color="Green">E243</font></i>: 
( <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A</font> is <a href="connsp_1.html#V2">connected</a> )
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T14" target="_self">Th14</a>, <a class="ref" href="rcomp_3.html#T16" target="_self">Th16</a>, <a class="ref" href="rcomp_3.html#T4" target="_self">Th4</a>, <a class="txt" href="rcomp_3.html#E39:83_1_1_1_2"><i><font color="Green">E121</font></i></a>, <a class="ref" href="rcomp_3.html#T31" target="_self">Th31</a>, <a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_15_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1_1_1_2_15_1"/>
(  ex <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">a</font></span><a href="rcomp_2.html#K1">.[</a></span> ) or  ex <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> ) or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K2">.[</a></span> ) )
 </span><b>by </b><i><a class="txt" href="rcomp_3.html#E43:83_1_1_1_2"><i><font color="Green">E125</font></i></a>, <a class="ref" href="rcomp_3.html#T17" target="_self">Th17</a>, <a class="txt" href="rcomp_3.html#E39:83_1_1_1_2"><i><font color="Green">E121</font></i></a>, <a class="txt" href="rcomp_3.html#E42:83_1_1_1_2"><i><font color="Green">E124</font></i></a>, <a class="ref" href="rcomp_3.html#T18" target="_self">Th18</a>, <a class="ref" href="rcomp_3.html#T30" target="_self">Th30</a>, <a class="txt" href="rcomp_3.html#E3:83_1_1_1_2"><i><font color="Green">E50</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_15_1_1"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_15_1_1"/>
 ex <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">a</font></span><a href="rcomp_2.html#K1">.[</a></span> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">a</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:83_1_1_1_2_15_1_1"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> )
 <b>and </b><br/><a NAME="E4:83_1_1_1_2_15_1_1"/><i><font color="Green">E244</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">a</font></span><a href="rcomp_2.html#K1">.[</a></span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_15_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1_1_1_2_15_1_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kL</font> or <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">kL</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_15_1_1_1_1"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_15_1_1_1_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kL</font>
 ;<br/><div class="add"><a NAME="E2:83_1_1_1_2_15_1_1_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">LEWY</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a>, <a class="ref" href="topreala.html#T12">TOPREALA:12</a></i>;<br/><a NAME="E3:83_1_1_1_2_15_1_1_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="rcomp_3.html#E40:83_1_1_1_2"><i><font color="Green">E122</font></i></a>, <a class="txt" href="rcomp_3.html#E39:83_1_1_1_2"><i><font color="Green">E121</font></i></a>, <a class="txt" href="rcomp_3.html#E19:83_1_1_1_2"><i><font color="Green">E102</font></i></a>, , <a class="ref" href="rcomp_3.html#T28" target="_self">Th28</a></i>;<br/><b>hence </b><a NAME="E4:83_1_1_1_2_15_1_1_1_1"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K2">.[</a></span> )
 <b>by </b><i><a class="txt" href="rcomp_3.html#E39:83_1_1_1_2"><i><font color="Green">E121</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_15_1_1_1_2"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_15_1_1_1_2"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">kL</font>
 ;<br/><div class="add"><a NAME="E2:83_1_1_1_2_15_1_1_1_2"/><b>then </b>
<font color="Maroon">LEWY</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a>, <a class="ref" href="topreala.html#T12">TOPREALA:12</a></i>;<br/><a NAME="E3:83_1_1_1_2_15_1_1_1_2"/><b>then </b>
<font color="Maroon">LEWY</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="rcomp_3.html#E40:83_1_1_1_2"><i><font color="Green">E122</font></i></a>, <a class="txt" href="rcomp_3.html#E39:83_1_1_1_2"><i><font color="Green">E121</font></i></a>, <a class="txt" href="rcomp_3.html#E19:83_1_1_1_2"><i><font color="Green">E102</font></i></a>, , <a class="ref" href="rcomp_3.html#T28" target="_self">Th28</a></i>;<br/><b>hence </b><a NAME="E4:83_1_1_1_2_15_1_1_1_2"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K2">.[</a></span> )
 <b>by </b><i><a class="txt" href="rcomp_3.html#E19:83_1_1_1_2"><i><font color="Green">E102</font></i></a>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_15_1_2"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_15_1_2"/>
 ex <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">a</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:83_1_1_1_2_15_1_2"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> )
 <b>and </b><br/><a NAME="E4:83_1_1_1_2_15_1_2"/><i><font color="Green">E245</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_15_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1_1_1_2_15_1_2_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">pP</font> or <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">pP</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_15_1_2_1_1"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_15_1_2_1_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">pP</font>
 ;<br/><div class="add"><a NAME="E2:83_1_1_1_2_15_1_2_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PRAWY</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a>, <a class="ref" href="topreala.html#T17">TOPREALA:17</a></i>;<br/><a NAME="E3:83_1_1_1_2_15_1_2_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="rcomp_3.html#E41:83_1_1_1_2"><i><font color="Green">E123</font></i></a>, <a class="txt" href="rcomp_3.html#E39:83_1_1_1_2"><i><font color="Green">E121</font></i></a>, <a class="ref" href="rcomp_3.html#T28" target="_self">Th28</a>, <a class="txt" href="rcomp_3.html#E29:83_1_1_1_2"><i><font color="Green">E114</font></i></a>, <a class="txt" href="rcomp_3.html#E25:83_1_1_1_2"><i><font color="Green">E112</font></i></a></i>;<br/><b>hence </b><a NAME="E4:83_1_1_1_2_15_1_2_1_1"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K2">.[</a></span> )
 <b>by </b><i><a class="txt" href="rcomp_3.html#E39:83_1_1_1_2"><i><font color="Green">E121</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_15_1_2_1_2"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_15_1_2_1_2"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">pP</font>
 ;<br/><div class="add"><a NAME="E2:83_1_1_1_2_15_1_2_1_2"/><b>then </b>
<font color="Maroon">PRAWY</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a>, <a class="ref" href="topreala.html#T17">TOPREALA:17</a></i>;<br/><a NAME="E3:83_1_1_1_2_15_1_2_1_2"/><b>then </b>
<font color="Maroon">PRAWY</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="rcomp_3.html#E41:83_1_1_1_2"><i><font color="Green">E123</font></i></a>, <a class="txt" href="rcomp_3.html#E39:83_1_1_1_2"><i><font color="Green">E121</font></i></a>, <a class="ref" href="rcomp_3.html#T28" target="_self">Th28</a>, <a class="txt" href="rcomp_3.html#E29:83_1_1_1_2"><i><font color="Green">E114</font></i></a>, <a class="txt" href="rcomp_3.html#E25:83_1_1_1_2"><i><font color="Green">E112</font></i></a></i>;<br/><b>hence </b><a NAME="E4:83_1_1_1_2_15_1_2_1_2"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K2">.[</a></span> )
 <b>by </b><i><a class="txt" href="rcomp_3.html#E29:83_1_1_1_2"><i><font color="Green">E114</font></i></a>, <a class="txt" href="rcomp_3.html#E25:83_1_1_1_2"><i><font color="Green">E112</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_15_1_3"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_15_1_3"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K2">.[</a></span> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">a</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:83_1_1_1_2_15_1_3"/><i><font color="Green">E246</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> )
 ;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font>, <font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>take </b>
<font color="Maroon">a</font>
;<br/><b>take </b>
<font color="Maroon">b</font>
;<br/><b>thus </b><a NAME="E5:83_1_1_1_2_15_1_3"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> )
 <b>by </b><i><a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E86:83_1_1_1_2"/><i><font color="Green">E247</font></i>: 
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  &amp;  <a href="pscomp_1.html#NK2" target="_self">sup</a> <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/><font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">PRAWY</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_16"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:83_1_1_1_2_16"/><i><font color="Green">E248</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> 
 <b>and </b><br/><a NAME="E2:83_1_1_1_2_16"/><i><font color="Green">E249</font></i>: 
 <a href="pscomp_1.html#NK2" target="_self">sup</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E3:83_1_1_1_2_16"/><i><font color="Green">E250</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">PRAWY</font>
 ;<br/>

<a NAME="E4:83_1_1_1_2_16"/>
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">LEWY</font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E41:83_1_1_1_2"><i><font color="Green">E123</font></i></a>, , <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:83_1_1_1_2_16"/><i><font color="Green">E251</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> )
 <b>by </b><i><a class="txt" href="rcomp_3.html#E40:83_1_1_1_2"><i><font color="Green">E122</font></i></a>, <a class="txt" href="rcomp_3.html#E42:83_1_1_1_2"><i><font color="Green">E124</font></i></a>, <a class="txt" href="rcomp_3.html#E37:83_1_1_1_2"><i><font color="Green">E120</font></i></a></i>;<br/>
<a NAME="E7:83_1_1_1_2_16"/>
 <a href="pscomp_1.html#NK2" target="_self">sup</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E43:83_1_1_1_2"><i><font color="Green">E125</font></i></a>, <a class="ref" href="topreal6.html#T22">TOPREAL6:22</a></i>;<br/>
<b>hence </b><a NAME="E8:83_1_1_1_2_16"/>
contradiction
 <b>by </b><i><a class="txt" href="rcomp_3.html#E41:83_1_1_1_2"><i><font color="Green">E123</font></i></a>, <a class="txt" href="rcomp_3.html#E43:83_1_1_1_2"><i><font color="Green">E125</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">S</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>3</sub></font> );<br/><a NAME="E87:83_1_1_1_2"/><i><font color="Green">E252</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> holds <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">X</font>  ex <font color="Olive">y</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">X</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_17"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:83_1_1_1_2_17"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> )
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">X</font>;<br/>

<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E3:83_1_1_1_2_17"/><i><font color="Green">E254</font></i>: 
not <font color="Maroon">x1</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E14:83_1_1_1_2"><i><font color="Green">E97</font></i></a></i>;<br/>
<a NAME="E4:83_1_1_1_2_17"/><i><font color="Green">E255</font></i>: 
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E5:83_1_1_1_2_17"/><b>then </b>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T17" target="_self">Th17</a>, <a class="ref" href="rcomp_3.html#T41" target="_self">Th41</a>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E6:83_1_1_1_2_17"/><b>then </b>
<font color="Maroon">x1</font> is <a href="seq_4.html#V1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T4">RCOMP_1:4</a></i>;<br/>
<a NAME="E7:83_1_1_1_2_17"/><b>then </b>
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="txt" href="rcomp_3.html#E41:83_1_1_1_2"><i><font color="Green">E123</font></i></a>, <a class="txt" href="rcomp_3.html#E42:83_1_1_1_2"><i><font color="Green">E124</font></i></a>, , <a class="ref" href="rcomp_3.html#T41" target="_self">Th41</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:83_1_1_1_2_17"/><i><font color="Green">E256</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E10:83_1_1_1_2_17"/><i><font color="Green">E257</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T41" target="_self">Th41</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">X</font> <b>by </b><i><a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>take </b>
<font color="Maroon">x</font>
;<br/>

<b>thus </b><a NAME="E12:83_1_1_1_2_17"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="txt" href="rcomp_3.html#E43:83_1_1_1_2"><i><font color="Green">E125</font></i></a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">LLEWY</font> = <font color="Maroon">LEWY</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">X</font> <b>by </b><i>, <a class="txt" href="rcomp_3.html#E34:83_1_1_1_2"><i><font color="Green">E118</font></i></a>, , <a class="ref" href="rcomp_3.html#T40" target="_self">Th40</a></i>;<br/><b>consider </b><font color="Maroon">IT</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E90:83_1_1_1_2"/><i><font color="Green">E260</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E91:83_1_1_1_2"/><i><font color="Green">E261</font></i>: 
( <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">LLEWY</font> or  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> 0 )
 <b>and </b><br/><a NAME="E92:83_1_1_1_2"/><i><font color="Green">E262</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>,<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>,<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>from </b><i><a class="ref" href="recdef_1.html#S6">RECDEF_1:sch 6</a>(<a class="txt" href="rcomp_3.html#E40:83_1_1_1_2"><i><font color="Green">E122</font></i></a>);<br/></i><a NAME="E93:83_1_1_1_2"/><i><font color="Green">E263</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">IT</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E94:83_1_1_1_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">IT</font> <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">IT</font> = <font color="Maroon">IT</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="pcomps_1.html#K1">bool</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><b>take </b>
<font color="Maroon">IT</font>
;<br/><b>thus </b><a NAME="E96:83_1_1_1_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">IT</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E13:83_1_1_1_2"><i><font color="Green">E96</font></i></a>, <a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E97:83_1_1_1_2"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/><a NAME="E98:83_1_1_1_2"/><b>then </b><i><font color="Green">E264</font></i>: 
not <font color="Maroon">IT</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E41:83_1_1_1_2"><i><font color="Green">E123</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E99:83_1_1_1_2"/><b>then </b><i><font color="Green">E265</font></i>: 
not  <a href="relset_1.html#K5">rng</a> <font color="Maroon">IT</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="finseq_1.html#T27">FINSEQ_1:27</a></i>;<br/><a NAME="E100:83_1_1_1_2"/><b>then </b><i><font color="Green">E266</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T34">FINSEQ_3:34</a></i>;<br/><a NAME="E101:83_1_1_1_2"/><b>then </b><i><font color="Green">E267</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E102:83_1_1_1_2"/><i><font color="Green">E268</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T44" target="_self">Th44</a>, <a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/><a NAME="E103:83_1_1_1_2"/><i><font color="Green">E269</font></i>: 
for <font color="Olive">n</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font> holds <br/>( <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_18"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:83_1_1_1_2_18"/><i><font color="Green">E270</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 ;<br/>

<a NAME="E2:83_1_1_1_2_18"/><i><font color="Green">E271</font></i>: 
<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T47" target="_self">Th47</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E3:83_1_1_1_2_18"/>
<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">IT</font>
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T47" target="_self">Th47</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:83_1_1_1_2_18"/>
( <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a>, <a class="ref" href="rcomp_3.html#T50" target="_self">Th50</a></i>;<br/>


</div><b>end;</b></div><a NAME="E104:83_1_1_1_2"/><i><font color="Green">E272</font></i>: 
for <font color="Olive">n</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font> holds <br/>( <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  &amp; <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_19"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:83_1_1_1_2_19"/><i><font color="Green">E273</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 ;<br/>

<a NAME="E2:83_1_1_1_2_19"/><i><font color="Green">E274</font></i>: 
<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T50" target="_self">Th50</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E3:83_1_1_1_2_19"/>
<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T50" target="_self">Th50</a>, <a class="ref" href="rcomp_3.html#T46" target="_self">Th46</a></i>;<br/>
<b>hence </b><a NAME="E4:83_1_1_1_2_19"/>
( <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  &amp; <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="rcomp_3.html#T51" target="_self">Th51</a>, <a class="ref" href="rcomp_3.html#T40" target="_self">Th40</a></i>;<br/>


</div><b>end;</b></div><a NAME="E105:83_1_1_1_2"/><i><font color="Green">E275</font></i>: 
for <font color="Olive">i</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font> holds <br/>for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>  st <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> holds <br/> ex <font color="Olive">j</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_20"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:83_1_1_1_2_20"/><i><font color="Green">E278</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[ <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font> implies for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>  st <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> holds <br/> ex <font color="Olive">j</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font> ) );<br/>
<a NAME="E2:83_1_1_1_2_20"/><i><font color="Green">E280</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<b>by </b><i><a class="ref" href="finseq_3.html#T26">FINSEQ_3:26</a></i>;<br/>
<a NAME="E3:83_1_1_1_2_20"/><i><font color="Green">E281</font></i>: 
for <font color="Olive">n</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:83_1_1_1_2_20_1"/><i><font color="Green">E282</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>]
 <b>and </b><br/><a NAME="E2:83_1_1_1_2_20_1"/><i><font color="Green">E283</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>;<br/>

<b>assume </b><a NAME="E3:83_1_1_1_2_20_1"/><i><font color="Green">E284</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<a NAME="E4:83_1_1_1_2_20_1"/><i><font color="Green">E285</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_20_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1_1_1_2_20_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font> )
 </span><b>by </b><i>, <a class="ref" href="topreala.html#T23">TOPREALA:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_20_1_1_1"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_20_1_1_1"/><i><font color="Green">E286</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>take </b>
1
;<br/><b>thus </b><a NAME="E2:83_1_1_1_2_20_1_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T34">FINSEQ_3:34</a></i>;<br/><b>thus </b><a NAME="E3:83_1_1_1_2_20_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="rcomp_3.html#T56" target="_self">Th56</a></i>;<br/><a NAME="E4:83_1_1_1_2_20_1_1_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>hence </b><a NAME="E5:83_1_1_1_2_20_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> 1
 <b>by </b><i><a class="ref" href="rcomp_3.html#T56" target="_self">Th56</a>, , , <a class="txt" href="rcomp_3.html#E42:83_1_1_1_2"><i><font color="Green">E124</font></i></a>, <a class="ref" href="rcomp_3.html#T55" target="_self">Th55</a>, <a class="ref" href="card_2.html#T59">CARD_2:59</a>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_20_1_1_2"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_20_1_1_2"/><i><font color="Green">E287</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 ;<br/><div class="add"><a NAME="E2:83_1_1_1_2_20_1_1_2"/><b>then </b><i><font color="Green">E288</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E3:83_1_1_1_2_20_1_1_2"/><i><font color="Green">E289</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T47" target="_self">Th47</a></i>;<br/><a NAME="E4:83_1_1_1_2_20_1_1_2"/><i><font color="Green">E290</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T46" target="_self">Th46</a></i>;<br/><a NAME="E5:83_1_1_1_2_20_1_1_2"/><i><font color="Green">E291</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> is  <a href="jct_misc.html#V1">connected</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="rcomp_3.html#T57" target="_self">Th57</a>, <a class="ref" href="rcomp_3.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E6:83_1_1_1_2_20_1_1_2"/><i><font color="Green">E292</font></i>: 
not <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T30" target="_self">Th30</a>, , <a class="ref" href="rcomp_3.html#T47" target="_self">Th47</a></i>;<br/><a NAME="E7:83_1_1_1_2_20_1_1_2"/><i><font color="Green">E293</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> is <a href="seq_4.html#V2">bounded_below</a>
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E8:83_1_1_1_2_20_1_1_2"/><i><font color="Green">E294</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T56" target="_self">Th56</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:83_1_1_1_2_20_1_1_2"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:83_1_1_1_2_20_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E11:83_1_1_1_2_20_1_1_2"/><b>then </b><i><font color="Green">E295</font></i>: 
 ex <font color="Olive">S</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> )
 <b>by </b><i>, <a class="txt" href="rcomp_3.html#E43:83_1_1_1_2"><i><font color="Green">E125</font></i></a>, <a class="ref" href="rcomp_3.html#T56" target="_self">Th56</a>, <a class="txt" href="rcomp_3.html#E41:83_1_1_1_2"><i><font color="Green">E123</font></i></a></i>;<br/><a NAME="E12:83_1_1_1_2_20_1_1_2"/><b>then </b><i><font color="Green">E296</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T55" target="_self">Th55</a>, , <a class="ref" href="rcomp_3.html#T59" target="_self">Th59</a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_20_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1_1_1_2_20_1_1_2_1"/>
( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> or <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_20_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_20_1_1_2_1_1"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">j</font> being  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:83_1_1_1_2_20_1_1_2_1_1"/><i><font color="Green">E297</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 <b>and </b><br/><a NAME="E4:83_1_1_1_2_20_1_1_2_1_1"/><i><font color="Green">E298</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E5:83_1_1_1_2_20_1_1_2_1_1"/><i><font color="Green">E299</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T56" target="_self">Th56</a></i>;<br/><b>take </b>
<font color="Maroon">j</font>
;<br/><b>thus </b><a NAME="E6:83_1_1_1_2_20_1_1_2_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 <b>by </b><i/>;<br/><a NAME="E7:83_1_1_1_2_20_1_1_2_1_1"/>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E8:83_1_1_1_2_20_1_1_2_1_1"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><b>thus </b><a NAME="E9:83_1_1_1_2_20_1_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_20_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_20_1_1_2_1_2"/><i><font color="Green">E300</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
;<br/><b>thus </b><a NAME="E2:83_1_1_1_2_20_1_1_2_1_2"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E3:83_1_1_1_2_20_1_1_2_1_2"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><b>thus </b><a NAME="E4:83_1_1_1_2_20_1_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T55" target="_self">Th55</a>, <a class="ref" href="rcomp_3.html#T56" target="_self">Th56</a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_20_1_1_2_1_3"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_20_1_1_2_1_3"/><i><font color="Green">E301</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
;<br/><b>thus </b><a NAME="E2:83_1_1_1_2_20_1_1_2_1_3"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E3:83_1_1_1_2_20_1_1_2_1_3"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><a NAME="E4:83_1_1_1_2_20_1_1_2_1_3"/>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:83_1_1_1_2_20_1_1_2_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="rcomp_3.html#T58" target="_self">Th58</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:83_1_1_1_2_20"/>
for <font color="Olive">n</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(<a class="ref" href="rcomp_3.html#T52" target="_self">Th52</a>, <a class="ref" href="rcomp_3.html#T54" target="_self">Th54</a>);<br/></i>
<b>hence </b><a NAME="E5:83_1_1_1_2_20"/>
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>  st <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> holds <br/> ex <font color="Olive">j</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font> )
 <b>by </b><i><a class="ref" href="rcomp_3.html#T51" target="_self">Th51</a></i>;<br/>


</div><b>end;</b></div><a NAME="E106:83_1_1_1_2"/><i><font color="Green">E302</font></i>: 
for <font color="Olive">i</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font> holds <br/>for <font color="Olive">j</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">j</font> holds <br/> ex <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font> &amp; ( for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> holds <br/><font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">y</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_21"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:83_1_1_1_2_21"/><i><font color="Green">E303</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[ <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a<sub>1</sub></font> implies  ex <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> holds <br/><font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">y</font> ) ) );<br/>
<a NAME="E2:83_1_1_1_2_21"/><i><font color="Green">E305</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
;<br/>
<a NAME="E3:83_1_1_1_2_21"/><i><font color="Green">E306</font></i>: 
for <font color="Olive">n</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_21_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:83_1_1_1_2_21_1"/><i><font color="Green">E307</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>]
 <b>and </b><br/><a NAME="E2:83_1_1_1_2_21_1"/><i><font color="Green">E308</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 ;<br/>

<a NAME="E3:83_1_1_1_2_21_1"/><i><font color="Green">E309</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>assume </b><a NAME="E4:83_1_1_1_2_21_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/>

<a NAME="E5:83_1_1_1_2_21_1"/><b>then </b><i><font color="Green">E310</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_21_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1_1_1_2_21_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font> )
 </span><b>by </b><i>, <a class="ref" href="topreala.html#T23">TOPREALA:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_21_1_1_1"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_21_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:83_1_1_1_2_21_1_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="rcomp_3.html#T56" target="_self">Th56</a></i>;<br/><b>hence </b><a NAME="E3:83_1_1_1_2_21_1_1_1"/>
 ex <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; ( for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> holds <br/><font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">y</font> ) )
 <b>by </b><i><a class="ref" href="rcomp_3.html#T52" target="_self">Th52</a>, <a class="ref" href="finseq_3.html#T26">FINSEQ_3:26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_21_1_1_2"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_21_1_1_2"/><i><font color="Green">E311</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 ;<br/><div class="add"><a NAME="E2:83_1_1_1_2_21_1_1_2"/><b>then </b><i><font color="Green">E312</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E3:83_1_1_1_2_21_1_1_2"/><b>then </b><i><font color="Green">E313</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">IT</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:83_1_1_1_2_21_1_1_2"/><i><font color="Green">E314</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T46" target="_self">Th46</a></i>;<br/><a NAME="E5:83_1_1_1_2_21_1_1_2"/><b>then </b><i><font color="Green">E315</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> is <a href="seq_4.html#V1">bounded_above</a>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E6:83_1_1_1_2_21_1_1_2"/><i><font color="Green">E316</font></i>: 
not <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, <a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="rcomp_3.html#E14:83_1_1_1_2"><i><font color="Green">E97</font></i></a></i>;<br/><a NAME="E7:83_1_1_1_2_21_1_1_2"/><i><font color="Green">E317</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="rcomp_3.html#E14:83_1_1_1_2"><i><font color="Green">E97</font></i></a></i>;<br/><a NAME="E8:83_1_1_1_2_21_1_1_2"/><i><font color="Green">E318</font></i>: 
<font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E9:83_1_1_1_2_21_1_1_2"/><i><font color="Green">E319</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:83_1_1_1_2_21_1_1_2"/><i><font color="Green">E320</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:83_1_1_1_2_21_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E41:83_1_1_1_2"><i><font color="Green">E123</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">S</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E13:83_1_1_1_2_21_1_1_2"/><i><font color="Green">E321</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E14:83_1_1_1_2_21_1_1_2"/><i><font color="Green">E322</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E43:83_1_1_1_2"><i><font color="Green">E125</font></i></a>, , </i>;<br/><a NAME="E15:83_1_1_1_2_21_1_1_2"/><i><font color="Green">E323</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">IT</font>
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T55" target="_self">Th55</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E16:83_1_1_1_2_21_1_1_2"/><i><font color="Green">E324</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T46" target="_self">Th46</a></i>;<br/><a NAME="E17:83_1_1_1_2_21_1_1_2"/><i><font color="Green">E325</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> is  <a href="pre_topc.html#V3">open</a> <a href="connsp_1.html#V2">connected</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="rcomp_3.html#E14:83_1_1_1_2"><i><font color="Green">E97</font></i></a></i>;<br/><a NAME="E18:83_1_1_1_2_21_1_1_2"/><b>then </b><i><font color="Green">E326</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> is  <a href="jct_misc.html#V1">connected</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="txt" href="rcomp_3.html#E2:83_1_1_1_2"><i><font color="Green">E49</font></i></a></i>;<br/><a NAME="E19:83_1_1_1_2_21_1_1_2"/><i><font color="Green">E327</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> is <a href="seq_4.html#V2">bounded_below</a>
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E20:83_1_1_1_2_21_1_1_2"/><i><font color="Green">E328</font></i>: 
not <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, <a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="rcomp_3.html#E14:83_1_1_1_2"><i><font color="Green">E97</font></i></a></i>;<br/><a NAME="E21:83_1_1_1_2_21_1_1_2"/><i><font color="Green">E329</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> is <a href="seq_4.html#V1">bounded_above</a>
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E22:83_1_1_1_2_21_1_1_2"/><b>then </b><i><font color="Green">E330</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">S</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T55" target="_self">Th55</a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/><a NAME="E23:83_1_1_1_2_21_1_1_2"/><i><font color="Green">E331</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T55" target="_self">Th55</a>, , <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><a NAME="E24:83_1_1_1_2_21_1_1_2"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="txt" href="rcomp_3.html#E47:83_1_1_1_2"><i><font color="Green">E160</font></i></a>, <a class="txt" href="rcomp_3.html#E48:83_1_1_1_2"><i><font color="Green">E161</font></i></a>, </i>;<br/><a NAME="E25:83_1_1_1_2_21_1_1_2"/><b>then </b><i><font color="Green">E332</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E26:83_1_1_1_2_21_1_1_2"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T59" target="_self">Th59</a>, <a class="ref" href="rcomp_3.html#T58" target="_self">Th58</a>, , , <a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a></i>;<br/><a NAME="E27:83_1_1_1_2_21_1_1_2"/><b>then </b><i><font color="Green">E333</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_21_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1_1_1_2_21_1_1_2_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 </span><b>by </b><i><a class="ref" href="rcomp_3.html#T56" target="_self">Th56</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_21_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_21_1_1_2_1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">y</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span><b> such that </b><br/><a NAME="E3:83_1_1_1_2_21_1_1_2_1_1"/><i><font color="Green">E334</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E4:83_1_1_1_2_21_1_1_2_1_1"/><i><font color="Green">E335</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> holds <br/><font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:83_1_1_1_2_21_1_1_2_1_1"/><i><font color="Green">E336</font></i>: 
<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T60" target="_self">Th60</a>, <a class="ref" href="rcomp_3.html#T58" target="_self">Th58</a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_21_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1_1_1_2_21_1_1_2_1_1_1"/>
(  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">S</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> or  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_21_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_21_1_1_2_1_1_1_1"/><i><font color="Green">E337</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">S</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
 ;<br/><div class="add"><b>set </b><font color="Maroon">y1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">S</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2;<br/><a NAME="E2:83_1_1_1_2_21_1_1_2_1_1_1_1"/><i><font color="Green">E339</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">S</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">S</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, <a class="ref" href="topreal3.html#T3">TOPREAL3:3</a></i>;<br/><a NAME="E3:83_1_1_1_2_21_1_1_2_1_1_1_1"/><i><font color="Green">E340</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">S</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="topreal3.html#T3">TOPREAL3:3</a></i>;<br/><a NAME="E4:83_1_1_1_2_21_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E341</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">S</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E50:83_1_1_1_2"><i><font color="Green">E164</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:83_1_1_1_2_21_1_1_2_1_1_1_1"/>
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, , <a class="ref" href="rcomp_3.html#T57" target="_self">Th57</a>, , <a class="ref" href="rcomp_3.html#T58" target="_self">Th58</a>, <a class="ref" href="rcomp_3.html#T59" target="_self">Th59</a>, </i>;<br/><a NAME="E6:83_1_1_1_2_21_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E7:83_1_1_1_2_21_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">S</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">y1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">S</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> <b>by </b><i><a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>take </b>
<font color="Maroon">y1</font>
;<br/><a NAME="E9:83_1_1_1_2_21_1_1_2_1_1_1_1"/>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">y1</font>
 <b>by </b><i>, <a class="txt" href="rcomp_3.html#E49:83_1_1_1_2"><i><font color="Green">E163</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E10:83_1_1_1_2_21_1_1_2_1_1_1_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="rcomp_3.html#E47:83_1_1_1_2"><i><font color="Green">E160</font></i></a>, <a class="txt" href="rcomp_3.html#E65"><i><font color="Green">Lemma171</font></i></a>, </i>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>;<br/><b>assume </b><a NAME="E11:83_1_1_1_2_21_1_1_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
 ;<br/><a NAME="E12:83_1_1_1_2_21_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/><a NAME="E13:83_1_1_1_2_21_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#D1" target="_self">Def1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E14:83_1_1_1_2_21_1_1_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T57" target="_self">Th57</a>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_21_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_21_1_1_2_1_1_1_2"/>
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:83_1_1_1_2_21_1_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E342</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">PRAWY</font>
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T55" target="_self">Th55</a>, , <a class="ref" href="rcomp_3.html#T40" target="_self">Th40</a>, <a class="txt" href="rcomp_3.html#E39:83_1_1_1_2"><i><font color="Green">E121</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">s</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> <b>by </b><i><a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="rcomp_3.html#T17" target="_self">Th17</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>take </b>
<font color="Maroon">y1</font>
;<br/><b>thus </b><a NAME="E4:83_1_1_1_2_21_1_1_2_1_1_1_2"/>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="rcomp_3.html#E28:83_1_1_1_2"><i><font color="Green">E113</font></i></a>, <a class="txt" href="rcomp_3.html#E29:83_1_1_1_2"><i><font color="Green">E114</font></i></a>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>;<br/><b>assume </b><a NAME="E5:83_1_1_1_2_21_1_1_2_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
 ;<br/><a NAME="E6:83_1_1_1_2_21_1_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/><a NAME="E7:83_1_1_1_2_21_1_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#D1" target="_self">Def1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E8:83_1_1_1_2_21_1_1_2_1_1_1_2"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E51:83_1_1_1_2"><i><font color="Green">E165</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_21_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_21_1_1_2_1_2"/><i><font color="Green">E343</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">y1</font> =  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> <b>by </b><i><a class="ref" href="rcomp_3.html#T59" target="_self">Th59</a>, <a class="ref" href="rcomp_3.html#T58" target="_self">Th58</a>, , , <a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a></i>;<br/><b>take </b>
<font color="Maroon">y1</font>
;<br/><b>thus </b><a NAME="E3:83_1_1_1_2_21_1_1_2_1_2"/>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T55" target="_self">Th55</a>, , , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>;<br/><b>assume </b><a NAME="E4:83_1_1_1_2_21_1_1_2_1_2"/><i><font color="Green">E344</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
 ;<br/><a NAME="E5:83_1_1_1_2_21_1_1_2_1_2"/><b>then </b><i><font color="Green">E345</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T60" target="_self">Th60</a>, <a class="ref" href="rcomp_3.html#T58" target="_self">Th58</a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_21_1_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_1_1_2_21_1_1_2_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E2:83_1_1_1_2_21_1_1_2_1_2_1"/><b>then </b><i><font color="Green">E346</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">PRAWY</font>
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T40" target="_self">Th40</a>, <a class="txt" href="rcomp_3.html#E39:83_1_1_1_2"><i><font color="Green">E121</font></i></a>, , <a class="ref" href="rcomp_3.html#T60" target="_self">Th60</a></i>;<br/><b>set </b><font color="Maroon">IT1</font> = <font color="Maroon">IT</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>;<br/><a NAME="E3:83_1_1_1_2_21_1_1_2_1_2_1"/><i><font color="Green">E348</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">IT</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/><a NAME="E4:83_1_1_1_2_21_1_1_2_1_2_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_21_1_1_2_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:83_1_1_1_2_21_1_1_2_1_2_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:83_1_1_1_2_21_1_1_2_1_2_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">IT</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:83_1_1_1_2_21_1_1_2_1_2_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:83_1_1_1_2_21_1_1_2_1_2_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">FI</font> =  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span> as   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> ;<br/><div><i><font color="Green">E350</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_21_1_1_2_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_1_1_2_21_1_1_2_1_2_1_2"/><i><font color="Green">E351</font></i>: 
<font color="Maroon">FI</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:83_1_1_1_2_21_1_1_2_1_2_1_2"/><i><font color="Green">E352</font></i>: 
<font color="Maroon">FI</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E3:83_1_1_1_2_21_1_1_2_1_2_1_2"/><b>then </b><i><font color="Green">E353</font></i>: 
<font color="Maroon">FI</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_3.html#T40" target="_self">Th40</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E4:83_1_1_1_2_21_1_1_2_1_2_1_2"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">FI</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_21_1_1_2_1_2_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:83_1_1_1_2_21_1_1_2_1_2_1_2_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">FI</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">l</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:83_1_1_1_2_21_1_1_2_1_2_1_2_1"/>
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">l</font> = <font color="Maroon">l</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_21_1_1_2_1_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1_1_1_2_21_1_1_2_1_2_1_2_1_1"/>
( <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> or <font color="Maroon">l</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_21_1_1_2_1_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_21_1_1_2_1_2_1_2_1_1_1"/>
<font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">j</font> being  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:83_1_1_1_2_21_1_1_2_1_2_1_2_1_1_1"/><i><font color="Green">E355</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 <b>and </b><br/><a NAME="E4:83_1_1_1_2_21_1_1_2_1_2_1_2_1_1_1"/><i><font color="Green">E356</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E5:83_1_1_1_2_21_1_1_2_1_2_1_2_1_1_1"/><i><font color="Green">E357</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T50" target="_self">Th50</a></i>;<br/><a NAME="E6:83_1_1_1_2_21_1_1_2_1_2_1_2_1_1_1"/><i><font color="Green">E358</font></i>: 
<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E7:83_1_1_1_2_21_1_1_2_1_2_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:83_1_1_1_2_21_1_1_2_1_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E9:83_1_1_1_2_21_1_1_2_1_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FI</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T73">FUNCT_1:73</a></i>;<br/><b>hence </b><a NAME="E10:83_1_1_1_2_21_1_1_2_1_2_1_2_1_1_1"/>
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">FI</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_21_1_1_2_1_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_21_1_1_2_1_2_1_2_1_1_2"/><i><font color="Green">E359</font></i>: 
<font color="Maroon">l</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:83_1_1_1_2_21_1_1_2_1_2_1_2_1_1_2"/>
<font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E3:83_1_1_1_2_21_1_1_2_1_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E360</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="txt" href="rcomp_3.html#E32:83_1_1_1_2"><i><font color="Green">E116</font></i></a>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:83_1_1_1_2_21_1_1_2_1_2_1_2_1_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:83_1_1_1_2_21_1_1_2_1_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FI</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T73">FUNCT_1:73</a></i>;<br/><b>hence </b><a NAME="E6:83_1_1_1_2_21_1_1_2_1_2_1_2_1_1_2"/>
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">FI</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E29:83_1_1_1_2"><i><font color="Green">E114</font></i></a>, <a class="txt" href="rcomp_3.html#E31:83_1_1_1_2"><i><font color="Green">E115</font></i></a>, <a class="ref" href="rcomp_3.html#T57" target="_self">Th57</a>, , , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E5:83_1_1_1_2_21_1_1_2_1_2_1_2"/><b>then </b>
<font color="Maroon">FI</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E23"><i><font color="Green">Lemma66</font></i></a>, <a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>;<br/><a NAME="E6:83_1_1_1_2_21_1_1_2_1_2_1_2"/><b>then </b><i><font color="Green">E361</font></i>: 
<font color="Maroon">FI</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">C</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E7:83_1_1_1_2_21_1_1_2_1_2_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">FI</font>,<font color="Maroon">M</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="wellord2.html#K1">RelIncl</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">C</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span>  ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_3.html#T61" target="_self">Th61</a>, <a class="ref" href="rcomp_3.html#T37" target="_self">Th37</a>, <a class="ref" href="rcomp_3.html#T35" target="_self">Th35</a>, <a class="ref" href="rcomp_3.html#T38" target="_self">Th38</a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/><b>hence </b><a NAME="E8:83_1_1_1_2_21_1_1_2_1_2_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="rcomp_3.html#T36" target="_self">Th36</a>, <a class="ref" href="rcomp_3.html#T35" target="_self">Th35</a>, <a class="ref" href="rcomp_3.html#T38" target="_self">Th38</a>, <a class="ref" href="orders_1.html#D12">ORDERS_1:def 12</a></i>;<br/></div><b>end;</b></div><a NAME="E7:83_1_1_1_2_21_1_1_2_1_2_1"/><i><font color="Green">E362</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">IT</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">IT</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="card_2.html#T66">CARD_2:66</a></i>;<br/><a NAME="E8:83_1_1_1_2_21_1_1_2_1_2_1"/>
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_21_1_1_2_1_2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:83_1_1_1_2_21_1_1_2_1_2_1_3"/><i><font color="Green">E363</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:83_1_1_1_2_21_1_1_2_1_2_1_3"/><i><font color="Green">E364</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T61" target="_self">Th61</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:83_1_1_1_2_21_1_1_2_1_2_1_3"/><i><font color="Green">E365</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T61" target="_self">Th61</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:83_1_1_1_2_21_1_1_2_1_2_1_3"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E6:83_1_1_1_2_21_1_1_2_1_2_1_3"/><b>then </b>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:83_1_1_1_2_21_1_1_2_1_2_1_3"/><b>then </b>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E8:83_1_1_1_2_21_1_1_2_1_2_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:83_1_1_1_2_21_1_1_2_1_2_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E10:83_1_1_1_2_21_1_1_2_1_2_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">IT</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>;<br/><a NAME="E11:83_1_1_1_2_21_1_1_2_1_2_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 0
 <b>by </b><i><a class="txt" href="rcomp_3.html#E41:83_1_1_1_2"><i><font color="Green">E123</font></i></a>, , <a class="txt" href="rcomp_3.html#E65"><i><font color="Green">Lemma171</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E12:83_1_1_1_2_21_1_1_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:83_1_1_1_2_21_1_1_2_1_2"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#D1" target="_self">Def1</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:83_1_1_1_2_21"/>
for <font color="Olive">n</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(<a class="ref" href="rcomp_3.html#T54" target="_self">Th54</a>, );<br/></i>
<b>hence </b><a NAME="E5:83_1_1_1_2_21"/>
for <font color="Olive">j</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">j</font> holds <br/> ex <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font> &amp; ( for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> holds <br/><font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">y</font> ) )
 ;<br/>


</div><b>end;</b></div><a NAME="E107:83_1_1_1_2"/><i><font color="Green">E366</font></i>: 
<font color="Maroon">IT</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>,<font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><a NAME="E108:83_1_1_1_2"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E109:83_1_1_1_2"/><b>then </b><i><font color="Green">E367</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E41:83_1_1_1_2"><i><font color="Green">E123</font></i></a>, <a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>;<br/><a NAME="E110:83_1_1_1_2"/><i><font color="Green">E368</font></i>: 
<font color="Maroon">IT</font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_22"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">j</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:83_1_1_1_2_22"/><i><font color="Green">E369</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 <b>and </b><br/><a NAME="E2:83_1_1_1_2_22"/><i><font color="Green">E370</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 <b>and </b><br/><a NAME="E3:83_1_1_1_2_22"/><i><font color="Green">E371</font></i>: 
<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/>

<b>assume </b><a NAME="E4:83_1_1_1_2_22"/><i><font color="Green">E372</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font>
 ;<br/>

<a NAME="E5:83_1_1_1_2_22"/><i><font color="Green">E373</font></i>: 
( <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font>, <font color="Maroon">j</font> = <font color="Maroon">j</font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i>, , <a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_22_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1_1_1_2_22_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font> or <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">i</font> )
 </span><b>by </b><i><a class="ref" href="rcomp_3.html#T56" target="_self">Th56</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_22_1_1"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_22_1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:83_1_1_1_2_22_1_1"/><b>then </b>
 ex <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> &amp; ( for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> holds <br/><font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">y</font> ) )
 <b>by </b><i>, , <a class="ref" href="rcomp_3.html#T51" target="_self">Th51</a></i>;<br/><b>hence </b><a NAME="E3:83_1_1_1_2_22_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T55" target="_self">Th55</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_22_1_2"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_22_1_2"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><a NAME="E2:83_1_1_1_2_22_1_2"/><b>then </b>
 ex <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> &amp; ( for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> holds <br/><font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">y</font> ) )
 <b>by </b><i>, , <a class="ref" href="rcomp_3.html#T51" target="_self">Th51</a></i>;<br/><b>hence </b><a NAME="E3:83_1_1_1_2_22_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T55" target="_self">Th55</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E111:83_1_1_1_2"/><b>then </b><i><font color="Green">E374</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T54" target="_self">Th54</a>, <a class="ref" href="rcomp_3.html#T52" target="_self">Th52</a>, <a class="ref" href="finseq_4.html#T78">FINSEQ_4:78</a></i>;<br/><b>hence </b><a NAME="E112:83_1_1_1_2"/>
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">IT</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T17" target="_self">Th17</a>, <a class="ref" href="rcomp_3.html#T41" target="_self">Th41</a>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/><a NAME="E113:83_1_1_1_2"/><i><font color="Green">E375</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font> &amp; <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">j</font> holds <br/><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_23"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">j</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:83_1_1_1_2_23"/><i><font color="Green">E376</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 <b>and </b><br/><a NAME="E2:83_1_1_1_2_23"/><i><font color="Green">E377</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 <b>and </b><br/><a NAME="E3:83_1_1_1_2_23"/><i><font color="Green">E378</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font>
 ;<br/>

<a NAME="E4:83_1_1_1_2_23"/>
( <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T55" target="_self">Th55</a>, <a class="ref" href="rcomp_3.html#T56" target="_self">Th56</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:83_1_1_1_2_23"/>
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T55" target="_self">Th55</a>, <a class="ref" href="rcomp_3.html#T56" target="_self">Th56</a>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E114:83_1_1_1_2"/>
 ex <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font> &amp; <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="rcomp_3.html#T41" target="_self">Th41</a>, <a class="ref" href="rcomp_3.html#T20" target="_self">Th20</a>, <a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E115:83_1_1_1_2"/><b>then </b>
<font color="Maroon">PRAWY</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T40" target="_self">Th40</a>, <a class="txt" href="rcomp_3.html#E36:83_1_1_1_2"><i><font color="Green">E119</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E117:83_1_1_1_2"/><i><font color="Green">E379</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 <b>and </b><br/><a NAME="E118:83_1_1_1_2"/><i><font color="Green">E380</font></i>: 
<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">PRAWY</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="rcomp_3.html#T55" target="_self">Th55</a></i>;<br/><a NAME="E120:83_1_1_1_2"/><i><font color="Green">E381</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T55" target="_self">Th55</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E121:83_1_1_1_2"/><i><font color="Green">E382</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T55" target="_self">Th55</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E122:83_1_1_1_2"/><i><font color="Green">E383</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T55" target="_self">Th55</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><div><i><font color="Green">E384</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_24"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:83_1_1_1_2_24"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font>
 ;<br/><a NAME="E2:83_1_1_1_2_24"/><b>then </b><i><font color="Green">E385</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T57" target="_self">Th57</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:83_1_1_1_2_24"/><b>then </b><i><font color="Green">E386</font></i>: 
 ex <font color="Olive">S</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="rcomp_3.html#E41:83_1_1_1_2"><i><font color="Green">E123</font></i></a>, <a class="txt" href="rcomp_3.html#E43:83_1_1_1_2"><i><font color="Green">E125</font></i></a>, </i>;<br/><a NAME="E4:83_1_1_1_2_24"/><i><font color="Green">E387</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E5:83_1_1_1_2_24"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T58" target="_self">Th58</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:83_1_1_1_2_24"/><b>then </b><i><font color="Green">E388</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:83_1_1_1_2_24"/><b>then </b><i><font color="Green">E389</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E8:83_1_1_1_2_24"/>
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T47" target="_self">Th47</a></i>;<br/><a NAME="E9:83_1_1_1_2_24"/><b>then </b>
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">PRAWY</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E32:83_1_1_1_2"><i><font color="Green">E116</font></i></a>, <a class="ref" href="rcomp_3.html#T56" target="_self">Th56</a>, <a class="ref" href="rcomp_3.html#T59" target="_self">Th59</a>, , <a class="txt" href="rcomp_3.html#E36:83_1_1_1_2"><i><font color="Green">E119</font></i></a></i>;<br/><a NAME="E10:83_1_1_1_2_24"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="rcomp_3.html#T56" target="_self">Th56</a>, , , <a class="ref" href="rcomp_3.html#T55" target="_self">Th55</a>, </i>;<br/><b>hence </b><a NAME="E11:83_1_1_1_2_24"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E124:83_1_1_1_2"/><i><font color="Green">E390</font></i>: 
for <font color="Olive">n</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font> holds <br/> ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K2">.[</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_25"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:83_1_1_1_2_25"/><i><font color="Green">E391</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E2:83_1_1_1_2_25"/><i><font color="Green">E392</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font>
 ;<br/>

<a NAME="E3:83_1_1_1_2_25"/><i><font color="Green">E393</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T59" target="_self">Th59</a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E4:83_1_1_1_2_25"/><b>then </b>
<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">IT</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:83_1_1_1_2_25"/><b>then </b><i><font color="Green">E394</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">IT</font>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E6:83_1_1_1_2_25"/><b>then </b><i><font color="Green">E395</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a></i>;<br/>
<a NAME="E7:83_1_1_1_2_25"/><b>then </b><i><font color="Green">E396</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_3.html#T40" target="_self">Th40</a></i>;<br/>
<a NAME="E8:83_1_1_1_2_25"/><i><font color="Green">E397</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="txt" href="rcomp_3.html#E13:83_1_1_1_2"><i><font color="Green">E96</font></i></a></i>;<br/>
<a NAME="E9:83_1_1_1_2_25"/><i><font color="Green">E398</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> is  <a href="pre_topc.html#V3">open</a> <a href="connsp_1.html#V2">connected</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="rcomp_3.html#E14:83_1_1_1_2"><i><font color="Green">E97</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_25_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1_1_1_2_25_1"/>
(  ex <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">a</font></span><a href="rcomp_2.html#K1">.[</a></span> ) or  ex <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> ) or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K2">.[</a></span> ) )
 </span><b>by </b><i>, <a class="ref" href="rcomp_3.html#T17" target="_self">Th17</a>, , , <a class="ref" href="rcomp_3.html#T18" target="_self">Th18</a>, <a class="ref" href="rcomp_3.html#T30" target="_self">Th30</a>, <a class="txt" href="rcomp_3.html#E3:83_1_1_1_2"><i><font color="Green">E50</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_25_1_1"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_25_1_1"/>
 ex <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">a</font></span><a href="rcomp_2.html#K1">.[</a></span> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">a</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:83_1_1_1_2_25_1_1"/><i><font color="Green">E399</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E4:83_1_1_1_2_25_1_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E5:83_1_1_1_2_25_1_1"/><i><font color="Green">E400</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">a</font></span><a href="rcomp_2.html#K1">.[</a></span>
 ;<br/><a NAME="E6:83_1_1_1_2_25_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">a</font></span><a href="rcomp_2.html#K1">.[</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/><a NAME="E7:83_1_1_1_2_25_1_1"/><b>then </b>
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">LEWY</font>
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T40" target="_self">Th40</a>, , <a class="txt" href="rcomp_3.html#E34:83_1_1_1_2"><i><font color="Green">E118</font></i></a></i>;<br/><b>hence </b><a NAME="E8:83_1_1_1_2_25_1_1"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K2">.[</a></span> )
 <b>by </b><i><a class="ref" href="rcomp_3.html#T59" target="_self">Th59</a>, , , <a class="txt" href="rcomp_3.html#E42:83_1_1_1_2"><i><font color="Green">E124</font></i></a>, , , <a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_25_1_2"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_25_1_2"/>
 ex <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">a</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:83_1_1_1_2_25_1_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E4:83_1_1_1_2_25_1_2"/><i><font color="Green">E401</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E5:83_1_1_1_2_25_1_2"/><i><font color="Green">E402</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span>
 ;<br/><a NAME="E6:83_1_1_1_2_25_1_2"/><i><font color="Green">E403</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:83_1_1_1_2_25_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/><a NAME="E8:83_1_1_1_2_25_1_2"/><b>then </b>
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">PRAWY</font>
 <b>by </b><i>, , , <a class="txt" href="rcomp_3.html#E39:83_1_1_1_2"><i><font color="Green">E121</font></i></a>, <a class="ref" href="rcomp_3.html#T40" target="_self">Th40</a></i>;<br/><b>hence </b><a NAME="E9:83_1_1_1_2_25_1_2"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K2">.[</a></span> )
 <b>by </b><i>, , , , , <a class="ref" href="rcomp_3.html#T56" target="_self">Th56</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_25_1_3"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_25_1_3"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.html#K2">.[</a></span> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">a</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:83_1_1_1_2_25_1_3"/><i><font color="Green">E404</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> )
 ;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font>, <font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>take </b>
<font color="Maroon">a</font>
;<br/><b>take </b>
<font color="Maroon">b</font>
;<br/><b>thus </b><a NAME="E5:83_1_1_1_2_25_1_3"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><div><i><font color="Green">E405</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_26"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:83_1_1_1_2_26"/><i><font color="Green">E406</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:83_1_1_1_2_26_1"/><i><font color="Green">E407</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font>
 ;<br/><a NAME="E2:83_1_1_1_2_26_1"/><b>then </b><i><font color="Green">E408</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:83_1_1_1_2_26_1"/><b>then </b>
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E4:83_1_1_1_2_26_1"/><b>then </b>
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">IT</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:83_1_1_1_2_26_1"/><b>then </b>
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a></i>;<br/><b>hence </b><a NAME="E6:83_1_1_1_2_26_1"/>
not <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T30" target="_self">Th30</a>, <a class="ref" href="rcomp_3.html#T40" target="_self">Th40</a></i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:83_1_1_1_2_26_2"/><i><font color="Green">E409</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font>
 ;<br/><a NAME="E2:83_1_1_1_2_26_2"/><b>then </b><i><font color="Green">E410</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:83_1_1_1_2_26_2"/><b>then </b><i><font color="Green">E411</font></i>: 
 ex <font color="Olive">S</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="rcomp_3.html#E41:83_1_1_1_2"><i><font color="Green">E123</font></i></a>, <a class="txt" href="rcomp_3.html#E43:83_1_1_1_2"><i><font color="Green">E125</font></i></a>, </i>;<br/><a NAME="E4:83_1_1_1_2_26_2"/><i><font color="Green">E412</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E5:83_1_1_1_2_26_2"/><i><font color="Green">E413</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:83_1_1_1_2_26_2"/><b>then </b><i><font color="Green">E414</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E7:83_1_1_1_2_26_2"/><i><font color="Green">E415</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:83_1_1_1_2_26_2"/><b>then </b>
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">IT</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E9:83_1_1_1_2_26_2"/><b>then </b><i><font color="Green">E416</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a></i>;<br/><a NAME="E10:83_1_1_1_2_26_2"/><b>then </b><i><font color="Green">E417</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_3.html#T40" target="_self">Th40</a></i>;<br/><a NAME="E11:83_1_1_1_2_26_2"/><i><font color="Green">E418</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E12:83_1_1_1_2_26_2"/><b>then </b><i><font color="Green">E419</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">IT</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E13:83_1_1_1_2_26_2"/><b>then </b><i><font color="Green">E420</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a></i>;<br/><a NAME="E14:83_1_1_1_2_26_2"/><b>then </b><i><font color="Green">E421</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_3.html#T40" target="_self">Th40</a></i>;<br/><a NAME="E15:83_1_1_1_2_26_2"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E16:83_1_1_1_2_26_2"/><b>then </b><i><font color="Green">E422</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E17:83_1_1_1_2_26_2"/><i><font color="Green">E423</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="seq_4.html#V3">bounded</a> <a href="jct_misc.html#V1">connected</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T40" target="_self">Th40</a>, <a class="ref" href="rcomp_3.html#T32" target="_self">Th32</a>, <a class="ref" href="rcomp_3.html#T30" target="_self">Th30</a>, <a class="ref" href="rcomp_3.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E18:83_1_1_1_2_26_2"/><i><font color="Green">E424</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> is  non <a href="xboole_0.html#V1">empty</a> <a href="seq_4.html#V3">bounded</a> <a href="jct_misc.html#V1">connected</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="txt" href="rcomp_3.html#E48:83_1_1_1_2"><i><font color="Green">E161</font></i></a>, <a class="ref" href="rcomp_3.html#T40" target="_self">Th40</a>, <a class="ref" href="rcomp_3.html#T32" target="_self">Th32</a>, <a class="ref" href="rcomp_3.html#T30" target="_self">Th30</a>, <a class="ref" href="rcomp_3.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E19:83_1_1_1_2_26_2"/>
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E47:83_1_1_1_2"><i><font color="Green">E160</font></i></a>, <a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E20:83_1_1_1_2_26_2"/><b>then </b>
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T17" target="_self">Th17</a>, <a class="ref" href="rcomp_3.html#T41" target="_self">Th41</a>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/><a NAME="E21:83_1_1_1_2_26_2"/><b>then </b><i><font color="Green">E425</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> is <a href="seq_4.html#V1">bounded_above</a>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T4">RCOMP_1:4</a></i>;<br/><a NAME="E22:83_1_1_1_2_26_2"/><b>then </b><i><font color="Green">E426</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:83_1_1_1_2_26_2_1"/><i><font color="Green">E427</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E2:83_1_1_1_2_26_2_1"/>
(  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> implies  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_26_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:83_1_1_1_2_26_2_1_1"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E2:83_1_1_1_2_26_2_1_1"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E3:83_1_1_1_2_26_2_1_1"/><b>then </b>
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">PRAWY</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_3.html#T40" target="_self">Th40</a>, <a class="txt" href="rcomp_3.html#E39:83_1_1_1_2"><i><font color="Green">E121</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:83_1_1_1_2_26_2_1_1"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i>, , , , , <a class="ref" href="rcomp_3.html#T56" target="_self">Th56</a>, </i>;<br/>


</div><b>end;</b></div><a NAME="E3:83_1_1_1_2_26_2_1"/><b>then </b>
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="rcomp_3.html#E51:83_1_1_1_2"><i><font color="Green">E165</font></i></a>, <a class="txt" href="rcomp_3.html#E50:83_1_1_1_2"><i><font color="Green">E164</font></i></a>, <a class="ref" href="rcomp_3.html#D1" target="_self">Def1</a>, </i>;<br/><a NAME="E4:83_1_1_1_2_26_2_1"/><b>then </b>
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> ) ) </span>}</span> 
 <b>by </b><i>, , <a class="ref" href="rcomp_3.html#T28" target="_self">Th28</a>, <a class="txt" href="rcomp_3.html#E49:83_1_1_1_2"><i><font color="Green">E163</font></i></a></i>;<br/><b>hence </b><a NAME="E5:83_1_1_1_2_26_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T40" target="_self">Th40</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div>
<b>end;</b></div><b>thus </b><a NAME="E24:83_1_1_1_2_26_2"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T60" target="_self">Th60</a>, , , , <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_26_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1_1_1_2_26_2_2"/>
( <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font> or <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_26_2_2_1"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_26_2_2_1"/><i><font color="Green">E428</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font>
 ;<br/><div class="add"><a NAME="E2:83_1_1_1_2_26_2_2_1"/><b>then </b>
<font color="Maroon">pP</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="rcomp_3.html#T56" target="_self">Th56</a>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/><b>hence </b><a NAME="E3:83_1_1_1_2_26_2_2_1"/>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#D1" target="_self">Def1</a>, , <a class="ref" href="rcomp_3.html#T56" target="_self">Th56</a>, , <a class="txt" href="rcomp_3.html#E24:83_1_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="txt" href="rcomp_3.html#E28:83_1_1_1_2"><i><font color="Green">E113</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_26_2_2_2"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_26_2_2_2"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">a1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:83_1_1_1_2_26_2_2_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font>
 <b>and </b><br/><a NAME="E4:83_1_1_1_2_26_2_2_2"/><i><font color="Green">E429</font></i>: 
<font color="Maroon">a1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b1</font>
 <b>and </b><br/><a NAME="E5:83_1_1_1_2_26_2_2_2"/>
<font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E6:83_1_1_1_2_26_2_2_2"/><i><font color="Green">E430</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T58" target="_self">Th58</a></i>;<br/><a NAME="E7:83_1_1_1_2_26_2_2_2"/>
<font color="Maroon">a1</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><b>hence </b><a NAME="E8:83_1_1_1_2_26_2_2_2"/>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#D1" target="_self">Def1</a>, , <a class="ref" href="topreal6.html#T22">TOPREAL6:22</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div></div><b>end;</b></div><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">n</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:83_1_1_1_2_27"/><i><font color="Green">E431</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><b>thus </b><a NAME="E2:83_1_1_1_2_27"/><i><font color="Green">E432</font></i>: 
( ( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font> implies  not <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> is <a href="xboole_0.html#V1">empty</a> ) &amp; ( <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font> implies (  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> ) ) )
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T59" target="_self">Th59</a></i>;<br/><b>assume </b><a NAME="E3:83_1_1_1_2_27"/><i><font color="Green">E433</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font>
 ;<br/><b>assume </b><a NAME="E4:83_1_1_1_2_27"/><i><font color="Green">E434</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">n</font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E6:83_1_1_1_2_27"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E7:83_1_1_1_2_27"/><b>then </b><i><font color="Green">E435</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T59" target="_self">Th59</a></i>;<br/><a NAME="E8:83_1_1_1_2_27"/><i><font color="Green">E436</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E9:83_1_1_1_2_27"/><b>then </b><i><font color="Green">E437</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:83_1_1_1_2_27"/><i><font color="Green">E438</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font>
 <b>by </b><i/>;<br/><a NAME="E11:83_1_1_1_2_27"/><b>then </b><i><font color="Green">E439</font></i>: 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">a1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E13:83_1_1_1_2_27"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font>
 <b>and </b><br/><a NAME="E14:83_1_1_1_2_27"/><i><font color="Green">E440</font></i>: 
<font color="Maroon">a1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b1</font>
 <b>and </b><br/><a NAME="E15:83_1_1_1_2_27"/>
<font color="Maroon">b1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E16:83_1_1_1_2_27"/><i><font color="Green">E441</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T58" target="_self">Th58</a></i>;<br/><a NAME="E17:83_1_1_1_2_27"/><i><font color="Green">E442</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font>
 <b>by </b><i>, <a class="ref" href="topreal6.html#T22">TOPREAL6:22</a></i>;<br/><a NAME="E18:83_1_1_1_2_27"/><i><font color="Green">E443</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font>
 <b>by </b><i>, <a class="ref" href="topreal6.html#T22">TOPREAL6:22</a></i>;<br/><a NAME="E19:83_1_1_1_2_27"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E20:83_1_1_1_2_27"/><b>then </b><i><font color="Green">E444</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E21:83_1_1_1_2_27"/><b>then </b>
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E22:83_1_1_1_2_27"/><b>then </b>
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">IT</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E23:83_1_1_1_2_27"/><b>then </b><i><font color="Green">E445</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a></i>;<br/><a NAME="E24:83_1_1_1_2_27"/><i><font color="Green">E446</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E25:83_1_1_1_2_27"/><b>then </b>
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E26:83_1_1_1_2_27"/><b>then </b>
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">IT</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E50:83_1_1_1_2"><i><font color="Green">E164</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E27:83_1_1_1_2_27"/><b>then </b><i><font color="Green">E447</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a></i>;<br/><a NAME="E28:83_1_1_1_2_27"/>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E29:83_1_1_1_2_27"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E30:83_1_1_1_2_27"/><b>then </b><i><font color="Green">E448</font></i>: 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E31:83_1_1_1_2_27"/><b>then </b>
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E32:83_1_1_1_2_27"/><b>then </b>
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">IT</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T60" target="_self">Th60</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E33:83_1_1_1_2_27"/><b>then </b><i><font color="Green">E449</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E44:83_1_1_1_2"><i><font color="Green">E126</font></i></a></i>;<br/><a NAME="E34:83_1_1_1_2_27"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E35:83_1_1_1_2_27"/><b>then </b><i><font color="Green">E450</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="rcomp_3.html#E50:83_1_1_1_2"><i><font color="Green">E164</font></i></a>, </i>;<br/><a NAME="E36:83_1_1_1_2_27"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E37:83_1_1_1_2_27"/><b>then </b><i><font color="Green">E451</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="rcomp_3.html#E50:83_1_1_1_2"><i><font color="Green">E164</font></i></a>, <a class="ref" href="rcomp_3.html#T60" target="_self">Th60</a></i>;<br/><a NAME="E38:83_1_1_1_2_27"/>
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_27_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:83_1_1_1_2_27_1"/><i><font color="Green">E452</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E3:83_1_1_1_2_27_1"/><i><font color="Green">E453</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b1</font> )
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_27_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1_1_1_2_27_1_1"/>
( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> or <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_27_1_1_1"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_27_1_1_1"/><i><font color="Green">E454</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_27_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1_1_1_2_27_1_1_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_27_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_27_1_1_1_1_1"/><i><font color="Green">E455</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:83_1_1_1_2_27_1_1_1_1_1"/><b>then </b>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="txt" href="rcomp_3.html#E42:83_1_1_1_2"><i><font color="Green">E124</font></i></a>, , <a class="txt" href="rcomp_3.html#E51:83_1_1_1_2"><i><font color="Green">E165</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a>, <a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/><a NAME="E3:83_1_1_1_2_27_1_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T61" target="_self">Th61</a>, , , <a class="txt" href="rcomp_3.html#E42:83_1_1_1_2"><i><font color="Green">E124</font></i></a>, , <a class="txt" href="rcomp_3.html#E18:83_1_1_1_2"><i><font color="Green">E101</font></i></a>, , <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a>, <a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/><b>hence </b><a NAME="E4:83_1_1_1_2_27_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_27_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_27_1_1_1_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:83_1_1_1_2_27_1_1_1_1_2"/><b>then </b><i><font color="Green">E456</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:83_1_1_1_2_27_1_1_1_1_2"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E4:83_1_1_1_2_27_1_1_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:83_1_1_1_2_27_1_1_1_1_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E7:83_1_1_1_2_27_1_1_1_1_2"/><i><font color="Green">E457</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E8:83_1_1_1_2_27_1_1_1_1_2"/>
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E9:83_1_1_1_2_27_1_1_1_1_2"/><i><font color="Green">E458</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T61" target="_self">Th61</a>, <a class="ref" href="rcomp_3.html#T58" target="_self">Th58</a></i>;<br/><a NAME="E10:83_1_1_1_2_27_1_1_1_1_2"/><i><font color="Green">E459</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="topreal6.html#T22">TOPREAL6:22</a></i>;<br/><a NAME="E11:83_1_1_1_2_27_1_1_1_1_2"/><i><font color="Green">E460</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="topreal6.html#T22">TOPREAL6:22</a></i>;<br/><a NAME="E12:83_1_1_1_2_27_1_1_1_1_2"/>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="txt" href="rcomp_3.html#E48:83_1_1_1_2"><i><font color="Green">E161</font></i></a>, <a class="txt" href="rcomp_3.html#E65"><i><font color="Green">Lemma171</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E13:83_1_1_1_2_27_1_1_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><b>hence </b><a NAME="E14:83_1_1_1_2_27_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_27_1_1_2"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_27_1_1_2"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:83_1_1_1_2_27_1_1_2"/><b>then </b><i><font color="Green">E461</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_27_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1_1_1_2_27_1_1_2_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 2 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_27_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_27_1_1_2_1_1"/><i><font color="Green">E462</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 2
 ;<br/><div class="add"><a NAME="E2:83_1_1_1_2_27_1_1_2_1_1"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, , <a class="txt" href="rcomp_3.html#E51:83_1_1_1_2"><i><font color="Green">E165</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E3:83_1_1_1_2_27_1_1_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E33:83_1_1_1_2"><i><font color="Green">E117</font></i></a>, , <a class="ref" href="rcomp_3.html#T61" target="_self">Th61</a>, , <a class="ref" href="rcomp_3.html#T56" target="_self">Th56</a>, , <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/><b>hence </b><a NAME="E4:83_1_1_1_2_27_1_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1_2_27_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:83_1_1_1_2_27_1_1_2_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 2
 ;<br/><div class="add"><a NAME="E2:83_1_1_1_2_27_1_1_2_1_2"/><b>then </b><i><font color="Green">E463</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:83_1_1_1_2_27_1_1_2_1_2"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E4:83_1_1_1_2_27_1_1_2_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon">a2</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:83_1_1_1_2_27_1_1_2_1_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font>
 <b>and </b><br/><a NAME="E7:83_1_1_1_2_27_1_1_2_1_2"/><i><font color="Green">E466</font></i>: 
<font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b2</font>
 <b>and </b><br/><a NAME="E8:83_1_1_1_2_27_1_1_2_1_2"/>
<font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E9:83_1_1_1_2_27_1_1_2_1_2"/><i><font color="Green">E467</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T61" target="_self">Th61</a>, <a class="ref" href="rcomp_3.html#T58" target="_self">Th58</a></i>;<br/><a NAME="E10:83_1_1_1_2_27_1_1_2_1_2"/><i><font color="Green">E468</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font>
 <b>by </b><i>, <a class="ref" href="topreal6.html#T22">TOPREAL6:22</a></i>;<br/><a NAME="E11:83_1_1_1_2_27_1_1_2_1_2"/><i><font color="Green">E469</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font>
 <b>by </b><i>, <a class="ref" href="topreal6.html#T22">TOPREAL6:22</a></i>;<br/><a NAME="E12:83_1_1_1_2_27_1_1_2_1_2"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b2</font>
 <b>by </b><i>, <a class="txt" href="rcomp_3.html#E65"><i><font color="Green">Lemma171</font></i></a>, , <a class="txt" href="rcomp_3.html#E47:83_1_1_1_2"><i><font color="Green">E160</font></i></a>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:83_1_1_1_2_27_1_1_2_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span>
 <b>by </b><i>, , , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><b>hence </b><a NAME="E14:83_1_1_1_2_27_1_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E39:83_1_1_1_2_27"/>
contradiction
 <b>by </b><i><a class="txt" href="rcomp_3.html#E49:83_1_1_1_2"><i><font color="Green">E163</font></i></a>, <a class="txt" href="rcomp_3.html#E51:83_1_1_1_2"><i><font color="Green">E165</font></i></a>, , <a class="ref" href="rcomp_3.html#D1" target="_self">Def1</a>, , <a class="ref" href="rcomp_3.html#T36" target="_self">Th36</a>, <a class="ref" href="rcomp_3.html#T37" target="_self">Th37</a>, <a class="ref" href="rcomp_3.html#T38" target="_self">Th38</a>, <a class="ref" href="rcomp_3.html#T7" target="_self">Th7</a></i>;<br/></div>
<b>end;</b></div><b>thus </b><a NAME="E127:83_1_1_1_2"/>
( <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> implies <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="binarith.html#K11">*&gt;</a></span> )
 <b>by </b><i><a class="ref" href="rcomp_3.html#T18" target="_self">Th18</a></i>;<br/><b>assume </b><a NAME="E128:83_1_1_1_2"/>
not <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><b>thus </b><a NAME="E129:83_1_1_1_2"/>
 ex <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Olive">p</font></span><a href="rcomp_2.html#K1">.[</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_28"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">kL</font>
;<br/>

<b>thus </b><a NAME="E1:83_1_1_1_2_28"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">kL</font>
 <b>by </b><i>, <a class="txt" href="rcomp_3.html#E18:83_1_1_1_2"><i><font color="Green">E101</font></i></a>, </i>;<br/>

<a NAME="E2:83_1_1_1_2_28"/>
 <a href="pscomp_1.html#NK2" target="_self">sup</a> <font color="Maroon">LEWY</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="seq_4.html#T65">SEQ_4:65</a></i>;<br/>
<b>hence </b><a NAME="E3:83_1_1_1_2_28"/>
<font color="Maroon">kL</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T17" target="_self">Th17</a>, <a class="ref" href="jordan5a.html#T20">JORDAN5A:20</a></i>;<br/>

<b>thus </b><a NAME="E4:83_1_1_1_2_28"/>
<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">kL</font></span><a href="rcomp_2.html#K1">.[</a></span>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E42:83_1_1_1_2"><i><font color="Green">E124</font></i></a>, <a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E130:83_1_1_1_2"/>
 ex <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Olive">p</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1_2_29"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">pP</font>
;<br/>

<a NAME="E1:83_1_1_1_2_29"/>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#NK3" target="_self">inf</a> <font color="Maroon">PRAWY</font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E22"><i><font color="Green">Lemma65</font></i></a>, <a class="ref" href="seq_4.html#T64">SEQ_4:64</a></i>;<br/>
<b>hence </b><a NAME="E2:83_1_1_1_2_29"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pP</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E33:83_1_1_1_2"><i><font color="Green">E117</font></i></a>, <a class="ref" href="rcomp_3.html#T17" target="_self">Th17</a>, <a class="ref" href="jordan5a.html#T20">JORDAN5A:20</a></i>;<br/>

<b>thus </b><a NAME="E3:83_1_1_1_2_29"/>
<font color="Maroon">pP</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E28:83_1_1_1_2"><i><font color="Green">E113</font></i></a>, <a class="txt" href="rcomp_3.html#E24:83_1_1_1_2"><i><font color="Green">E111</font></i></a>, </i>;<br/>

<b>thus </b><a NAME="E4:83_1_1_1_2_29"/>
<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">pP</font>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T56" target="_self">Th56</a>, </i>;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon">n</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E131:83_1_1_1_2"/><i><font color="Green">E470</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font> )
 ;<br/><b>consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E133:83_1_1_1_2"/><i><font color="Green">E471</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E134:83_1_1_1_2"/><i><font color="Green">E472</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E135:83_1_1_1_2"/><i><font color="Green">E473</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E136:83_1_1_1_2"/><i><font color="Green">E474</font></i>: 
<font color="Maroon">IT</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T58" target="_self">Th58</a></i>;<br/><b>take </b>
<font color="Maroon">a</font>
;<br/><b>take </b>
<font color="Maroon">b</font>
;<br/><a NAME="E137:83_1_1_1_2"/>
<font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E138:83_1_1_1_2"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> )
 <b>by </b><i>, , , , , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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