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

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

<b>assume </b><b>that </b><br/><a NAME="E1:54"/><i><font color="Green">E33</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">X</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E2:54"/><i><font color="Green">E34</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">Y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E3:54"/><i><font color="Green">E35</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">Y</font> &amp;  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> implies  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>and </b><br/><a NAME="E4:54"/><i><font color="Green">E45</font></i>: 
(  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">Y</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> implies  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 ;<br/>

<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="E5:54"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<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="E7:54"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">Y</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">Y</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="topreal6.html#T24">TOPREAL6:24</a></i>;<br/>
<a NAME="E8:54"/><b>then </b><i><font color="Green">E47</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">Y</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>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E4:54"><i><font color="Green">E45</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E9:54"/><b>then </b><i><font color="Green">E48</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">X</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>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">X</font> )
 
<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="54_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_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="54_1_1"><b>suppose </b></a><a NAME="E1:54_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="E2:54_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_2"><b>suppose </b></a><a NAME="E1:54_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> 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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_3"><b>suppose </b></a><a NAME="E1:54_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="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="E3:54_1_3"/><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> &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="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K1">.]</a></span> )
 ;<br/><a NAME="E4:54_1_3"/>
(  <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="txt" href="rcomp_3.html#E9:54"><i><font color="Green">E48</font></i></a>, <a class="ref" href="jordan5a.html#T20">JORDAN5A:20</a></i>;<br/><b>hence </b><a NAME="E5:54_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E9:54"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rcomp_3.html#E8:54"><i><font color="Green">E47</font></i></a>, <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="54_1_4"><b>suppose </b></a><a NAME="E1:54_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">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="E3:54_1_4"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">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="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_2.html#K1">.[</a></span> )
 ;<br/><a NAME="E4:54_1_4"/><i><font color="Green">E51</font></i>: 
(  <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="txt" href="rcomp_3.html#E9:54"><i><font color="Green">E48</font></i></a>, , </i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1_4_1"/>
( <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="54_1_4_1_1"><b>suppose </b></a><a NAME="E1:54_1_4_1_1"/>
<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="E2:54_1_4_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_4_1_2"><b>suppose </b></a><a NAME="E1:54_1_4_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">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="E2:54_1_4_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_4_1_3"><b>suppose </b></a><a NAME="E1:54_1_4_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="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">a1</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b1</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="E3:54_1_4_1_3"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b1</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="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="rcomp_1.html#K1">.]</a></span> )
 ;<br/><a NAME="E4:54_1_4_1_3"/><i><font color="Green">E53</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> )
 <b>by </b><i><a class="txt" href="rcomp_3.html#E3:54_1_4"><i><font color="Green">E50</font></i></a>, <a class="ref" href="jordan5a.html#T20">JORDAN5A:20</a></i>;<br/><a NAME="E5:54_1_4_1_3"/><b>then </b>
<font color="Maroon">b1</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#E9:54"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rcomp_3.html#E3:54_1_4"><i><font color="Green">E50</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E6:54_1_4_1_3"/><b>then </b>
<font color="Maroon">x</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#E5:54"><i><font color="Green">E46</font></i></a>, <a class="txt" href="rcomp_3.html#E4:54_1_4"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E7:54_1_4_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E9:54"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rcomp_3.html#E8:54"><i><font color="Green">E47</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="54_1_4_1_4"><b>suppose </b></a><a NAME="E1:54_1_4_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_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">a1</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b1</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="E3:54_1_4_1_4"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b1</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="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="rcomp_2.html#K1">.[</a></span> )
 ;<br/><a NAME="E4:54_1_4_1_4"/><i><font color="Green">E95</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> )
 <b>by </b><i><a class="txt" href="rcomp_3.html#E3:54_1_4"><i><font color="Green">E50</font></i></a>, , </i>;<br/><a NAME="E5:54_1_4_1_4"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b1</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E4:54"><i><font color="Green">E45</font></i></a>, <a class="txt" href="rcomp_3.html#E3:54_1_4"><i><font color="Green">E50</font></i></a>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/><a NAME="E6:54_1_4_1_4"/><b>then </b>
<font color="Maroon">x</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#E4:54_1_4"><i><font color="Green">E51</font></i></a>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E7:54_1_4_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E9:54"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rcomp_3.html#E8:54"><i><font color="Green">E47</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="54_1_4_1_5"><b>suppose </b></a><a NAME="E1:54_1_4_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#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">a1</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b1</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="E3:54_1_4_1_5"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b1</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="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="rcomp_2.html#K2">.]</a></span> )
 ;<br/><a NAME="E4:54_1_4_1_5"/><i><font color="Green">E97</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> )
 <b>by </b><i><a class="txt" href="rcomp_3.html#E3:54_1_4"><i><font color="Green">E50</font></i></a>, , </i>;<br/><a NAME="E5:54_1_4_1_5"/><b>then </b>
<font color="Maroon">b1</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#E9:54"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rcomp_3.html#E3:54_1_4"><i><font color="Green">E50</font></i></a>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E6:54_1_4_1_5"/><b>then </b>
<font color="Maroon">x</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#E5:54"><i><font color="Green">E46</font></i></a>, <a class="txt" href="rcomp_3.html#E4:54_1_4"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E7:54_1_4_1_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E9:54"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rcomp_3.html#E8:54"><i><font color="Green">E47</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="54_1_4_1_6"><b>suppose </b></a><a NAME="E1:54_1_4_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_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">a1</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b1</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="E3:54_1_4_1_6"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b1</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="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="rcomp_1.html#K2">.[</a></span> )
 ;<br/><a NAME="E4:54_1_4_1_6"/><i><font color="Green">E99</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> )
 <b>by </b><i><a class="txt" href="rcomp_3.html#E3:54_1_4"><i><font color="Green">E50</font></i></a>, <a class="ref" href="topreal6.html#T22">TOPREAL6:22</a></i>;<br/><a NAME="E5:54_1_4_1_6"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b1</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E4:54"><i><font color="Green">E45</font></i></a>, <a class="txt" href="rcomp_3.html#E3:54_1_4"><i><font color="Green">E50</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E6:54_1_4_1_6"/><b>then </b>
<font color="Maroon">x</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#E4:54_1_4"><i><font color="Green">E51</font></i></a>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E7:54_1_4_1_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E9:54"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rcomp_3.html#E8:54"><i><font color="Green">E47</font></i></a>, , <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</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="54_1_5"><b>suppose </b></a><a NAME="E1:54_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">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="E3:54_1_5"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">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="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_2.html#K2">.]</a></span> )
 ;<br/><a NAME="E4:54_1_5"/><i><font color="Green">E101</font></i>: 
(  <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="txt" href="rcomp_3.html#E9:54"><i><font color="Green">E48</font></i></a>, , </i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1_5_1"/>
( <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="54_1_5_1_1"><b>suppose </b></a><a NAME="E1:54_1_5_1_1"/>
<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="E2:54_1_5_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_5_1_2"><b>suppose </b></a><a NAME="E1:54_1_5_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">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="E2:54_1_5_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_5_1_3"><b>suppose </b></a><a NAME="E1:54_1_5_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="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">a1</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b1</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="E3:54_1_5_1_3"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b1</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="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="rcomp_1.html#K1">.]</a></span> )
 ;<br/><a NAME="E4:54_1_5_1_3"/><i><font color="Green">E103</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> )
 <b>by </b><i>, <a class="ref" href="jordan5a.html#T20">JORDAN5A:20</a></i>;<br/><a NAME="E5:54_1_5_1_3"/><b>then </b>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a1</font>
 <b>by </b><i>, , , <a class="txt" href="rcomp_3.html#E9:54"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E6:54_1_5_1_3"/><b>then </b>
<font color="Maroon">a</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#E5:54"><i><font color="Green">E46</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E7:54_1_5_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E9:54"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rcomp_3.html#E8:54"><i><font color="Green">E47</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="54_1_5_1_4"><b>suppose </b></a><a NAME="E1:54_1_5_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_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">a1</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b1</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="E3:54_1_5_1_4"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b1</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="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="rcomp_2.html#K1">.[</a></span> )
 ;<br/><a NAME="E4:54_1_5_1_4"/>
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E5:54_1_5_1_4"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a1</font>
 <b>by </b><i>, , , <a class="txt" href="rcomp_3.html#E9:54"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E6:54_1_5_1_4"/>
<font color="Maroon">a1</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#E4:54"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/><a NAME="E7:54_1_5_1_4"/><b>then </b>
<font color="Maroon">a</font> <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="E8:54_1_5_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E9:54"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rcomp_3.html#E8:54"><i><font color="Green">E47</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="54_1_5_1_5"><b>suppose </b></a><a NAME="E1:54_1_5_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#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">a1</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b1</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="E3:54_1_5_1_5"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b1</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="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="rcomp_2.html#K2">.]</a></span> )
 ;<br/><a NAME="E4:54_1_5_1_5"/><i><font color="Green">E107</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E5:54_1_5_1_5"/>
<font color="Maroon">a1</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#E4:54"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/><a NAME="E6:54_1_5_1_5"/><b>then </b>
<font color="Maroon">a</font> <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="E7:54_1_5_1_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E9:54"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rcomp_3.html#E8:54"><i><font color="Green">E47</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="54_1_5_1_6"><b>suppose </b></a><a NAME="E1:54_1_5_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_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">a1</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b1</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="E3:54_1_5_1_6"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b1</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="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="rcomp_1.html#K2">.[</a></span> )
 ;<br/><a NAME="E4:54_1_5_1_6"/><i><font color="Green">E109</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">Y</font> <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="E5:54_1_5_1_6"/>
<font color="Maroon">a1</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#E4:54"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E6:54_1_5_1_6"/><b>then </b>
<font color="Maroon">a</font> <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="E7:54_1_5_1_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E9:54"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rcomp_3.html#E8:54"><i><font color="Green">E47</font></i></a>, , <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</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="54_1_6"><b>suppose </b></a><a NAME="E1:54_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#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="E3:54_1_6"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">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="Maroon">a</font>,<font color="Maroon">b</font></span><a href="rcomp_1.html#K2">.[</a></span> )
 ;<br/><a NAME="E4:54_1_6"/><i><font color="Green">E111</font></i>: 
(  <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="txt" href="rcomp_3.html#E9:54"><i><font color="Green">E48</font></i></a>, <a class="ref" href="topreal6.html#T22">TOPREAL6:22</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1_6_1"/>
( <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="54_1_6_1_1"><b>suppose </b></a><a NAME="E1:54_1_6_1_1"/>
<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="E2:54_1_6_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_6_1_2"><b>suppose </b></a><a NAME="E1:54_1_6_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">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="E2:54_1_6_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_6_1_3"><b>suppose </b></a><a NAME="E1:54_1_6_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="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">a1</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b1</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="E3:54_1_6_1_3"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b1</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="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="rcomp_1.html#K1">.]</a></span> )
 ;<br/><a NAME="E4:54_1_6_1_3"/><i><font color="Green">E113</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> )
 <b>by </b><i>, <a class="ref" href="jordan5a.html#T20">JORDAN5A:20</a></i>;<br/><a NAME="E5:54_1_6_1_3"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b1</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E4:54"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E6:54_1_6_1_3"/>
<font color="Maroon">b1</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#E9:54"><i><font color="Green">E48</font></i></a>, , , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E7:54_1_6_1_3"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E8:54_1_6_1_3"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a1</font>
 <b>by </b><i>, , , , <a class="txt" href="rcomp_3.html#E9:54"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E9:54_1_6_1_3"/><b>then </b>
<font color="Maroon">a</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#E5:54"><i><font color="Green">E46</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E10:54_1_6_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E9:54"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_6_1_4"><b>suppose </b></a><a NAME="E1:54_1_6_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_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">a1</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b1</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="E3:54_1_6_1_4"/><i><font color="Green">E116</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b1</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="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="rcomp_2.html#K1">.[</a></span> )
 ;<br/><a NAME="E4:54_1_6_1_4"/><i><font color="Green">E117</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E5:54_1_6_1_4"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b1</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E4:54"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/><a NAME="E6:54_1_6_1_4"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font>
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:54_1_6_1_4"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a1</font>
 <b>by </b><i>, , , <a class="txt" href="rcomp_3.html#E9:54"><i><font color="Green">E48</font></i></a>, , , <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E8:54_1_6_1_4"/>
<font color="Maroon">a1</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#E4:54"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/><a NAME="E9:54_1_6_1_4"/><b>then </b>
<font color="Maroon">a</font> <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="E10:54_1_6_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E9:54"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_6_1_5"><b>suppose </b></a><a NAME="E1:54_1_6_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#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">a1</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b1</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="E3:54_1_6_1_5"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b1</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="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="rcomp_2.html#K2">.]</a></span> )
 ;<br/><a NAME="E4:54_1_6_1_5"/><i><font color="Green">E121</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E5:54_1_6_1_5"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b1</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E4:54"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/><a NAME="E6:54_1_6_1_5"/>
<font color="Maroon">b1</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#E9:54"><i><font color="Green">E48</font></i></a>, , , <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E7:54_1_6_1_5"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E8:54_1_6_1_5"/>
<font color="Maroon">a1</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#E4:54"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/><a NAME="E9:54_1_6_1_5"/><b>then </b>
<font color="Maroon">a</font> <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="E10:54_1_6_1_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E9:54"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_6_1_6"><b>suppose </b></a><a NAME="E1:54_1_6_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_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">a1</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b1</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="E3:54_1_6_1_6"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b1</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="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="rcomp_1.html#K2">.[</a></span> )
 ;<br/><a NAME="E4:54_1_6_1_6"/><i><font color="Green">E125</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">Y</font> <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="E5:54_1_6_1_6"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b1</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E4:54"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E6:54_1_6_1_6"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font>
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:54_1_6_1_6"/>
<font color="Maroon">a1</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#E4:54"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E8:54_1_6_1_6"/><b>then </b>
<font color="Maroon">a</font> <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="E9:54_1_6_1_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E9:54"><i><font color="Green">E48</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></div><b>end;</b></div>

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