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

<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</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> ;<br/>



<b>assume </b><a NAME="E1:53"/><i><font color="Green">E33</font></i>: 
not <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a>  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 <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Olive">a</font> 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 <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Olive">a</font> 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 <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Olive">a</font> 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 <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Olive">a</font> 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="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.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="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</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="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <font color="Maroon">X</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="Olive">b</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="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <font color="Maroon">X</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/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1"><b>suppose </b></a><a NAME="E1:53_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:53_1_1"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">X</font> or  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">X</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> )
 <b>by </b><i>, <a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_2"><b>suppose </b></a><a NAME="E1:53_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 <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Olive">a</font>
 ;<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="E3:53_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K2">left_closed_halfline</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E4:53_1_2"/>
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:53_1_2"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">X</font> or  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">X</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> )
 <b>by </b><i>, , <a class="ref" href="borsuk_5.html#T16">BORSUK_5:16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_3"><b>suppose </b></a><a NAME="E1:53_1_3"/>
 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 <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Olive">a</font>
 ;<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="E3:53_1_3"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E4:53_1_3"/>
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:53_1_3"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">X</font> or  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">X</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> )
 <b>by </b><i>, , <a class="ref" href="borsuk_5.html#T17">BORSUK_5:17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_4"><b>suppose </b></a><a NAME="E1:53_1_4"/>
 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 <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Olive">a</font>
 ;<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="E3:53_1_4"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K3">right_closed_halfline</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E4:53_1_4"/>
 <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:53_1_4"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">X</font> or  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">X</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> )
 <b>by </b><i>, , <a class="ref" href="borsuk_5.html#T15">BORSUK_5:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_5"><b>suppose </b></a><a NAME="E1:53_1_5"/>
 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 <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Olive">a</font>
 ;<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="E3:53_1_5"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E4:53_1_5"/>
 <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:53_1_5"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">X</font> or  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">X</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> )
 <b>by </b><i>, , <a class="ref" href="borsuk_5.html#T14">BORSUK_5:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_6"><b>suppose </b></a><a NAME="E1:53_1_6"/>
 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="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="rcomp_1.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> , <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="E3:53_1_6"/><i><font color="Green">E47</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="E4:53_1_6"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/><a NAME="E5:53_1_6"/>
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, , <a class="ref" href="jordan5a.html#T20">JORDAN5A:20</a></i>;<br/><b>hence </b><a NAME="E6:53_1_6"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">X</font> or  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">X</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> )
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_7"><b>suppose </b></a><a NAME="E1:53_1_7"/>
 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="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</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> , <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="E3:53_1_7"/><i><font color="Green">E49</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="E4:53_1_7"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_2.html#K1">.[</a></span>
 ;<br/><a NAME="E5:53_1_7"/>
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E6:53_1_7"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">X</font> or  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">X</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> )
 <b>by </b><i>, , <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="53_1_8"><b>suppose </b></a><a NAME="E1:53_1_8"/>
 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="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <font color="Maroon">X</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="Olive">b</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> , <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="E3:53_1_8"/><i><font color="Green">E51</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="E4:53_1_8"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">X</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">b</font></span><a href="rcomp_2.html#K2">.]</a></span>
 ;<br/><a NAME="E5:53_1_8"/>
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E6:53_1_8"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">X</font> or  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">X</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> )
 <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="53_1_9"><b>suppose </b></a><a NAME="E1:53_1_9"/>
 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="Olive">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b</font> &amp; <font color="Maroon">X</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="E3:53_1_9"/><i><font color="Green">E53</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="E4:53_1_9"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">X</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/><a NAME="E5:53_1_9"/>
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">X</font> <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/><b>hence </b><a NAME="E6:53_1_9"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">X</font> or  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">X</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> )
 <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>
