<?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">s</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<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/>
<b>assume </b><a NAME="E1:74"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 ;<br/>

<b>let </b><font color="Maroon">X</font> be  <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>;<br/>

<a NAME="E2:74"/><i><font color="Green">E34</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>  <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>
 
;<br/>
<a NAME="E3:74"/><i><font color="Green">E35</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="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E4:74"/><b>then </b>
<font color="Maroon">X</font> is  <a href="seq_4.html#V1">bounded_above</a> <a href="seq_4.html#V2">bounded_below</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_1.html#T3">RCOMP_1:3</a>, <a class="ref" href="rcomp_1.html#T4">RCOMP_1:4</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">X</font> as  <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/>;<br/>
<a NAME="E6:74"/><i><font color="Green">E45</font></i>: 
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> is <a href="connsp_1.html#V1">connected</a>
 
<b>by </b><i>, <a class="ref" href="treal_1.html#T23">TREAL_1:23</a></i>;<br/>
<a NAME="E7:74"/><i><font color="Green">E46</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_1.html#T15">RCOMP_1:15</a></i>;<br/>
<a NAME="E8:74"/><i><font color="Green">E47</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_1.html#T15">RCOMP_1:15</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_1"/>
( <font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a> or <font color="Maroon">Y</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">Y</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">Y</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">Y</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">Y</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">Y</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">Y</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">Y</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">Y</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="74_1_1"><b>suppose </b></a><a NAME="E1:74_1_1"/>
<font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:74_1_1"/>
( <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> or <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">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.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">X</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">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="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">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><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2"><b>suppose </b></a><a NAME="E1:74_1_2"/>
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:74_1_2"/>
( <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> or <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">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.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">X</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">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="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">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> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_3"><b>suppose </b></a><a NAME="E1:74_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">Y</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">Y</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">Y</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">Y</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>hence </b><a NAME="E2:74_1_3"/>
( <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> or <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">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.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">X</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">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="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">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><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_4"><b>suppose </b></a><a NAME="E1:74_1_4"/>
 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">Y</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="E3:74_1_4"/><i><font color="Green">E48</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="E4:74_1_4"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">Y</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="E5:74_1_4"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b</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#T12" target="_self">Th12</a>, <a class="ref" href="rcomp_3.html#T7" target="_self">Th7</a>, <a class="ref" href="borsuk_4.html#T14">BORSUK_4:14</a></i>;<br/><a NAME="E6:74_1_4"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</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#T7" target="_self">Th7</a>, <a class="ref" href="rcomp_3.html#T12" target="_self">Th12</a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><a NAME="E7:74_1_4"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">X</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T12" target="_self">Th12</a>, <a class="ref" href="rcomp_3.html#T14" target="_self">Th14</a>, <a class="ref" href="topreala.html#T44">TOPREALA:44</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:74_1_4_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> or <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">s</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_1_4_1_1"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font> or <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> )
 </span><b>by </b><i><a class="ref" href="rcomp_3.html#T14" target="_self">Th14</a>, <a class="ref" href="rcomp_3.html#T18" target="_self">Th18</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="74_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:74_1_4_1_1_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><a NAME="E2:74_1_4_1_1_1"/><b>then </b>
not <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T12" target="_self">Th12</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>hence </b><a NAME="E3:74_1_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rcomp_3.html#E1:74_1_4_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="rcomp_3.html#E7:74_1_4"><i><font color="Green">E52</font></i></a>, <a class="ref" href="rcomp_3.html#T5" target="_self">Th5</a>, <a class="ref" href="rcomp_3.html#T16" target="_self">Th16</a>, <a class="ref" href="rcomp_3.html#T17" target="_self">Th17</a>, <a class="ref" href="rcomp_3.html#T4" target="_self">Th4</a>, <a class="ref" href="connsp_1.html#T14">CONNSP_1:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:74_1_4_1_1_2"/>
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><a NAME="E2:74_1_4_1_1_2"/><b>then </b>
not <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="rcomp_3.html#T12" target="_self">Th12</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>hence </b><a NAME="E3:74_1_4_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="rcomp_3.html#E1:74_1_4_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="rcomp_3.html#E7:74_1_4"><i><font color="Green">E52</font></i></a>, <a class="ref" href="rcomp_3.html#T6" target="_self">Th6</a>, <a class="ref" href="rcomp_3.html#T16" target="_self">Th16</a>, <a class="ref" href="rcomp_3.html#T17" target="_self">Th17</a>, <a class="ref" href="rcomp_3.html#T4" target="_self">Th4</a>, <a class="ref" href="connsp_1.html#T14">CONNSP_1:14</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:74_1_4"/>
( <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> or <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">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.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">X</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">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="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">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> ) )
 <b>by </b><i><a class="ref" href="rcomp_3.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_5"><b>suppose </b></a><a NAME="E1:74_1_5"/>
 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">Y</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="E3:74_1_5"/><i><font color="Green">E91</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="E4:74_1_5"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">Y</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="E5:74_1_5"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b</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#T12" target="_self">Th12</a>, <a class="ref" href="rcomp_3.html#T7" target="_self">Th7</a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:74_1_5_1"/>
<font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E2:74_1_5_1"/><b>then </b><i><font color="Green">E97</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="ref" href="rcomp_3.html#T14" target="_self">Th14</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_5_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:74_1_5_1_1"/><i><font color="Green">E98</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:74_1_5_1_1"/>
 <a href="tops_1.html#K1">Int</a> <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>
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T12" target="_self">Th12</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="txt" href="rcomp_3.html#E8:74"><i><font color="Green">E47</font></i></a></i>;<br/><a NAME="E3:74_1_5_1_1"/><b>then </b>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><b>hence </b><a NAME="E4:74_1_5_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="rcomp_3.html#T17" target="_self">Th17</a>, <a class="ref" href="rcomp_3.html#T12" target="_self">Th12</a>, <a class="ref" href="rcomp_3.html#T7" target="_self">Th7</a>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:74_1_5_1"/>
contradiction
 <b>by </b><i><a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:74_1_5"/>
( <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> or <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">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.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">X</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">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="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">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> ) )
 <b>by </b><i><a class="ref" href="rcomp_3.html#T7" target="_self">Th7</a>, <a class="ref" href="rcomp_3.html#T12" target="_self">Th12</a>, <a class="ref" href="rcomp_3.html#T14" target="_self">Th14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_6"><b>suppose </b></a><a NAME="E1:74_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">Y</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="E3:74_1_6"/><i><font color="Green">E99</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="E4:74_1_6"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">Y</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="E5:74_1_6"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b</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#T12" target="_self">Th12</a>, <a class="ref" href="rcomp_3.html#T7" target="_self">Th7</a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:74_1_6_1"/>
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">s</font>
 ;<br/><a NAME="E2:74_1_6_1"/><b>then </b><i><font color="Green">E102</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="ref" href="rcomp_3.html#T14" target="_self">Th14</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_6_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:74_1_6_1_1"/><i><font color="Green">E103</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:74_1_6_1_1"/>
 <a href="tops_1.html#K1">Int</a> <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>
 <b>by </b><i>, <a class="ref" href="rcomp_3.html#T12" target="_self">Th12</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>, </i>;<br/><a NAME="E3:74_1_6_1_1"/><b>then </b>
not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><b>hence </b><a NAME="E4:74_1_6_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="rcomp_3.html#T17" target="_self">Th17</a>, <a class="ref" href="rcomp_3.html#T12" target="_self">Th12</a>, <a class="ref" href="rcomp_3.html#T7" target="_self">Th7</a>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:74_1_6_1"/>
contradiction
 <b>by </b><i><a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:74_1_6"/>
( <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> or <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">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.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">X</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">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="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">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> ) )
 <b>by </b><i><a class="ref" href="rcomp_3.html#T7" target="_self">Th7</a>, <a class="ref" href="rcomp_3.html#T12" target="_self">Th12</a>, <a class="ref" href="rcomp_3.html#T14" target="_self">Th14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_7"><b>suppose </b></a><a NAME="E1:74_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">Y</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="E3:74_1_7"/><i><font color="Green">E104</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="E4:74_1_7"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">Y</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="E5:74_1_7"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b</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#T12" target="_self">Th12</a>, <a class="ref" href="rcomp_3.html#T7" target="_self">Th7</a>, <a class="ref" href="borsuk_4.html#T15">BORSUK_4:15</a></i>;<br/><b>hence </b><a NAME="E6:74_1_7"/>
( <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> or <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">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.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">X</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">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="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">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> ) )
 <b>by </b><i><a class="ref" href="rcomp_3.html#T7" target="_self">Th7</a>, <a class="ref" href="rcomp_3.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
