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

<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a> <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">connected</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">Y</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">Y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">X</font> &amp; (  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">Y</font> &amp;  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> implies  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> ) &amp; (  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">Y</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> implies  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> ) implies <font color="Maroon" title="c2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:39"/><i><font color="Green" title="E34">A1</font></i>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">Y</font>
 <b>and </b><br/><a NAME="E2:39"/><i><font color="Green" title="E35">A2</font></i>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">Y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">X</font>
 <b>and </b><br/><a NAME="E3:39"/><i><font color="Green" title="E36">A3</font></i>: 
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">Y</font> &amp;  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> implies  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )
 <b>and </b><br/><a NAME="E4:39"/><i><font color="Green" title="E37">A4</font></i>: 
(  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">Y</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> implies  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font></span><br/>

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<b>assume </b><a NAME="E5:39"/><i><font color="Green" title="E38">A5</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c3">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E7:39"/>
<font color="Maroon" title="c2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="topreal6.html#T24" title="TOPREAL6:th.24">TOPREAL6:24</a></i>;<br/>
<a NAME="E8:39"/><b>then </b><i><font color="Green" title="E39">A6</font></i>: 
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">Y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">Y</font> )
 
<b>by </b><i><a class="txt" href="#E5:39"><i><font color="Green" title="E38">A5</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/>
<a NAME="E9:39"/><b>then </b><i><font color="Green" title="E40">A7</font></i>: 
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="#E2:39"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1"/>
( <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  or  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Olive" title="b1">a</font> or  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Olive" title="b1">a</font> or  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Olive" title="b1">a</font> or  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Olive" title="b1">a</font> or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> ) or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> ) or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> ) or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> ) )
 </span><b>by </b><i><a class="ref" href="rcomp_3.html#T41" target="_self" title="RCOMP_3:th.41">Th41</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1"><b>suppose </b></a><a NAME="E1:39_1_1"/>
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><b>hence </b><a NAME="E2:39_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2"><b>suppose </b></a><a NAME="E1:39_1_2"/>
(  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Olive" title="b1">a</font> or  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Olive" title="b1">a</font> or  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Olive" title="b1">a</font> or  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Olive" title="b1">a</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><b>hence </b><a NAME="E2:39_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_3"><b>suppose </b></a><a NAME="E1:39_1_3"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:39_1_3"/><i><font color="Green" title="E41">A8</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 ;<br/><a NAME="E4:39_1_3"/>
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="txt" href="#E3:39_1_3"><i><font color="Green" title="E41">A8</font></i></a>, <a class="ref" href="jordan5a.html#T20" title="JORDAN5A:th.20">JORDAN5A:20</a></i>;<br/><b>hence </b><a NAME="E5:39_1_3"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E9:39"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="#E3:39_1_3"><i><font color="Green" title="E41">A8</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_4"><b>suppose </b></a><a NAME="E1:39_1_4"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:39_1_4"/><i><font color="Green" title="E41">A9</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> )
 ;<br/><a NAME="E4:39_1_4"/><i><font color="Green" title="E42">A10</font></i>: 
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="txt" href="#E3:39_1_4"><i><font color="Green" title="E41">A9</font></i></a>, <a class="ref" href="rcomp_3.html#T4" target="_self" title="RCOMP_3:th.4">Th4</a>, <a class="ref" href="rcomp_3.html#T5" target="_self" title="RCOMP_3:th.5">Th5</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_4_1"/>
( <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  or  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Olive" title="b1">a</font> or  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Olive" title="b1">a</font> or  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Olive" title="b1">a</font> or  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Olive" title="b1">a</font> or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> ) or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> ) or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> ) or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> ) )
 </span><b>by </b><i><a class="ref" href="rcomp_3.html#T41" target="_self" title="RCOMP_3:th.41">Th41</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_4_1_1"><b>suppose </b></a><a NAME="E1:39_1_4_1_1"/>
<font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><b>hence </b><a NAME="E2:39_1_4_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_4_1_2"><b>suppose </b></a><a NAME="E1:39_1_4_1_2"/>
(  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Olive" title="b1">a</font> or  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Olive" title="b1">a</font> or  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Olive" title="b1">a</font> or  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Olive" title="b1">a</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><b>hence </b><a NAME="E2:39_1_4_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_4_1_3"><b>suppose </b></a><a NAME="E1:39_1_4_1_3"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c8">b1</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:39_1_4_1_3"/><i><font color="Green" title="E43">A11</font></i>: 
( <font color="Maroon" title="c7">a1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">b1</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 ;<br/><a NAME="E4:39_1_4_1_3"/><i><font color="Green" title="E44">A12</font></i>: 
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a1</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b1</font> )
 <b>by </b><i><a class="txt" href="#E3:39_1_4_1_3"><i><font color="Green" title="E43">A11</font></i></a>, <a class="ref" href="jordan5a.html#T20" title="JORDAN5A:th.20">JORDAN5A:20</a></i>;<br/><a NAME="E5:39_1_4_1_3"/><b>then </b>
<font color="Maroon" title="c8">b1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="#E2:39"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="#E4:39"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="#E3:39_1_4"><i><font color="Green" title="E41">A9</font></i></a>, <a class="txt" href="#E4:39_1_4"><i><font color="Green" title="E42">A10</font></i></a>, <a class="txt" href="#E3:39_1_4_1_3"><i><font color="Green" title="E43">A11</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</a></i>;<br/><a NAME="E6:39_1_4_1_3"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="#E8:39"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="#E4:39_1_4_1_3"><i><font color="Green" title="E44">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E7:39_1_4_1_3"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E9:39"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="#E3:39_1_4"><i><font color="Green" title="E41">A9</font></i></a>, <a class="txt" href="#E4:39_1_4"><i><font color="Green" title="E42">A10</font></i></a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_4_1_4"><b>suppose </b></a><a NAME="E1:39_1_4_1_4"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c8">b1</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:39_1_4_1_4"/><i><font color="Green" title="E43">A13</font></i>: 
( <font color="Maroon" title="c7">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> )
 ;<br/><a NAME="E4:39_1_4_1_4"/><i><font color="Green" title="E44">A14</font></i>: 
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a1</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b1</font> )
 <b>by </b><i><a class="txt" href="#E3:39_1_4_1_4"><i><font color="Green" title="E43">A13</font></i></a>, <a class="ref" href="rcomp_3.html#T4" target="_self" title="RCOMP_3:th.4">Th4</a>, <a class="ref" href="rcomp_3.html#T5" target="_self" title="RCOMP_3:th.5">Th5</a></i>;<br/><a NAME="E5:39_1_4_1_4"/>
<font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font>
 <b>by </b><i><a class="txt" href="#E5:39"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="#E3:39_1_4_1_4"><i><font color="Green" title="E43">A13</font></i></a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</a></i>;<br/><a NAME="E6:39_1_4_1_4"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="#E2:39"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="#E4:39_1_4"><i><font color="Green" title="E42">A10</font></i></a>, <a class="txt" href="#E4:39_1_4_1_4"><i><font color="Green" title="E44">A14</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E7:39_1_4_1_4"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E9:39"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="#E3:39_1_4"><i><font color="Green" title="E41">A9</font></i></a>, <a class="txt" href="#E4:39_1_4"><i><font color="Green" title="E42">A10</font></i></a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_4_1_5"><b>suppose </b></a><a NAME="E1:39_1_4_1_5"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c8">b1</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:39_1_4_1_5"/><i><font color="Green" title="E43">A15</font></i>: 
( <font color="Maroon" title="c7">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )
 ;<br/><a NAME="E4:39_1_4_1_5"/><i><font color="Green" title="E44">A16</font></i>: 
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a1</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b1</font> )
 <b>by </b><i><a class="txt" href="#E3:39_1_4_1_5"><i><font color="Green" title="E43">A15</font></i></a>, <a class="ref" href="rcomp_3.html#T6" target="_self" title="RCOMP_3:th.6">Th6</a>, <a class="ref" href="rcomp_3.html#T7" target="_self" title="RCOMP_3:th.7">Th7</a></i>;<br/><a NAME="E5:39_1_4_1_5"/><b>then </b>
<font color="Maroon" title="c8">b1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="#E2:39"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="#E4:39"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="#E3:39_1_4"><i><font color="Green" title="E41">A9</font></i></a>, <a class="txt" href="#E4:39_1_4"><i><font color="Green" title="E42">A10</font></i></a>, <a class="txt" href="#E3:39_1_4_1_5"><i><font color="Green" title="E43">A15</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a>, <a class="ref" href="xxreal_1.html#T2" title="XXREAL_1:th.2">XXREAL_1:2</a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</a></i>;<br/><a NAME="E6:39_1_4_1_5"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="#E8:39"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="#E4:39_1_4_1_5"><i><font color="Green" title="E44">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E7:39_1_4_1_5"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E9:39"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="#E3:39_1_4"><i><font color="Green" title="E41">A9</font></i></a>, <a class="txt" href="#E4:39_1_4"><i><font color="Green" title="E42">A10</font></i></a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_4_1_6"><b>suppose </b></a><a NAME="E1:39_1_4_1_6"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c8">b1</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:39_1_4_1_6"/><i><font color="Green" title="E43">A17</font></i>: 
( <font color="Maroon" title="c7">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 ;<br/><a NAME="E4:39_1_4_1_6"/><i><font color="Green" title="E44">A18</font></i>: 
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a1</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b1</font> )
 <b>by </b><i><a class="txt" href="#E3:39_1_4_1_6"><i><font color="Green" title="E43">A17</font></i></a>, <a class="ref" href="topreal6.html#T22" title="TOPREAL6:th.22">TOPREAL6:22</a></i>;<br/><a NAME="E5:39_1_4_1_6"/>
<font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font>
 <b>by </b><i><a class="txt" href="#E5:39"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="#E3:39_1_4_1_6"><i><font color="Green" title="E43">A17</font></i></a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></i>;<br/><a NAME="E6:39_1_4_1_6"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="#E2:39"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="#E4:39_1_4"><i><font color="Green" title="E42">A10</font></i></a>, <a class="txt" href="#E4:39_1_4_1_6"><i><font color="Green" title="E44">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E7:39_1_4_1_6"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E9:39"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="#E3:39_1_4"><i><font color="Green" title="E41">A9</font></i></a>, <a class="txt" href="#E4:39_1_4"><i><font color="Green" title="E42">A10</font></i></a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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="39_1_5"><b>suppose </b></a><a NAME="E1:39_1_5"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:39_1_5"/><i><font color="Green" title="E41">A19</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )
 ;<br/><a NAME="E4:39_1_5"/><i><font color="Green" title="E42">A20</font></i>: 
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="txt" href="#E3:39_1_5"><i><font color="Green" title="E41">A19</font></i></a>, <a class="ref" href="rcomp_3.html#T6" target="_self" title="RCOMP_3:th.6">Th6</a>, <a class="ref" href="rcomp_3.html#T7" target="_self" title="RCOMP_3:th.7">Th7</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_5_1"/>
( <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  or  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Olive" title="b1">a</font> or  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Olive" title="b1">a</font> or  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Olive" title="b1">a</font> or  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Olive" title="b1">a</font> or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> ) or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> ) or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> ) or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> ) )
 </span><b>by </b><i><a class="ref" href="rcomp_3.html#T41" target="_self" title="RCOMP_3:th.41">Th41</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_5_1_1"><b>suppose </b></a><a NAME="E1:39_1_5_1_1"/>
<font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><b>hence </b><a NAME="E2:39_1_5_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_5_1_2"><b>suppose </b></a><a NAME="E1:39_1_5_1_2"/>
(  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Olive" title="b1">a</font> or  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Olive" title="b1">a</font> or  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Olive" title="b1">a</font> or  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Olive" title="b1">a</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><b>hence </b><a NAME="E2:39_1_5_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_5_1_3"><b>suppose </b></a><a NAME="E1:39_1_5_1_3"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c8">b1</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:39_1_5_1_3"/><i><font color="Green" title="E43">A21</font></i>: 
( <font color="Maroon" title="c7">a1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">b1</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 ;<br/><a NAME="E4:39_1_5_1_3"/><i><font color="Green" title="E44">A22</font></i>: 
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a1</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b1</font> )
 <b>by </b><i><a class="txt" href="#E3:39_1_5_1_3"><i><font color="Green" title="E43">A21</font></i></a>, <a class="ref" href="jordan5a.html#T20" title="JORDAN5A:th.20">JORDAN5A:20</a></i>;<br/><a NAME="E5:39_1_5_1_3"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">a1</font>
 <b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="#E3:39"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="#E3:39_1_5"><i><font color="Green" title="E41">A19</font></i></a>, <a class="txt" href="#E4:39_1_5"><i><font color="Green" title="E42">A20</font></i></a>, <a class="txt" href="#E3:39_1_5_1_3"><i><font color="Green" title="E43">A21</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a>, <a class="ref" href="xxreal_1.html#T2" title="XXREAL_1:th.2">XXREAL_1:2</a></i>;<br/><a NAME="E6:39_1_5_1_3"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E8:39"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="#E4:39_1_5_1_3"><i><font color="Green" title="E44">A22</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E7:39_1_5_1_3"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E9:39"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="#E3:39_1_5"><i><font color="Green" title="E41">A19</font></i></a>, <a class="txt" href="#E4:39_1_5"><i><font color="Green" title="E42">A20</font></i></a>, <a class="ref" href="xxreal_1.html#T2" title="XXREAL_1:th.2">XXREAL_1:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_5_1_4"><b>suppose </b></a><a NAME="E1:39_1_5_1_4"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c8">b1</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:39_1_5_1_4"/><i><font color="Green" title="E43">A23</font></i>: 
( <font color="Maroon" title="c7">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> )
 ;<br/><a NAME="E4:39_1_5_1_4"/>
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a1</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b1</font> )
 <b>by </b><i><a class="txt" href="#E3:39_1_5_1_4"><i><font color="Green" title="E43">A23</font></i></a>, <a class="ref" href="rcomp_3.html#T4" target="_self" title="RCOMP_3:th.4">Th4</a>, <a class="ref" href="rcomp_3.html#T5" target="_self" title="RCOMP_3:th.5">Th5</a></i>;<br/><a NAME="E5:39_1_5_1_4"/><b>then </b><i><font color="Green" title="E44">A24</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">a1</font>
 <b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="#E3:39"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="#E3:39_1_5"><i><font color="Green" title="E41">A19</font></i></a>, <a class="txt" href="#E4:39_1_5"><i><font color="Green" title="E42">A20</font></i></a>, <a class="txt" href="#E3:39_1_5_1_4"><i><font color="Green" title="E43">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a>, <a class="ref" href="xxreal_1.html#T2" title="XXREAL_1:th.2">XXREAL_1:2</a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</a></i>;<br/><a NAME="E6:39_1_5_1_4"/>
<font color="Maroon" title="c7">a1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E5:39"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="#E3:39_1_5_1_4"><i><font color="Green" title="E43">A23</font></i></a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</a></i>;<br/><a NAME="E7:39_1_5_1_4"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E5:39_1_5_1_4"><i><font color="Green" title="E44">A24</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E8:39_1_5_1_4"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E9:39"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="#E3:39_1_5"><i><font color="Green" title="E41">A19</font></i></a>, <a class="txt" href="#E4:39_1_5"><i><font color="Green" title="E42">A20</font></i></a>, <a class="ref" href="xxreal_1.html#T2" title="XXREAL_1:th.2">XXREAL_1:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_5_1_5"><b>suppose </b></a><a NAME="E1:39_1_5_1_5"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c8">b1</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:39_1_5_1_5"/><i><font color="Green" title="E43">A25</font></i>: 
( <font color="Maroon" title="c7">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )
 ;<br/><a NAME="E4:39_1_5_1_5"/><i><font color="Green" title="E44">A26</font></i>: 
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a1</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b1</font> )
 <b>by </b><i><a class="txt" href="#E3:39_1_5_1_5"><i><font color="Green" title="E43">A25</font></i></a>, <a class="ref" href="rcomp_3.html#T6" target="_self" title="RCOMP_3:th.6">Th6</a>, <a class="ref" href="rcomp_3.html#T7" target="_self" title="RCOMP_3:th.7">Th7</a></i>;<br/><a NAME="E5:39_1_5_1_5"/>
<font color="Maroon" title="c7">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E5:39"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="#E3:39_1_5_1_5"><i><font color="Green" title="E43">A25</font></i></a>, <a class="ref" href="xxreal_1.html#T2" title="XXREAL_1:th.2">XXREAL_1:2</a></i>;<br/><a NAME="E6:39_1_5_1_5"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="#E4:39_1_5"><i><font color="Green" title="E42">A20</font></i></a>, <a class="txt" href="#E4:39_1_5_1_5"><i><font color="Green" title="E44">A26</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E7:39_1_5_1_5"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E9:39"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="#E3:39_1_5"><i><font color="Green" title="E41">A19</font></i></a>, <a class="txt" href="#E4:39_1_5"><i><font color="Green" title="E42">A20</font></i></a>, <a class="ref" href="xxreal_1.html#T2" title="XXREAL_1:th.2">XXREAL_1:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_5_1_6"><b>suppose </b></a><a NAME="E1:39_1_5_1_6"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c8">b1</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:39_1_5_1_6"/><i><font color="Green" title="E43">A27</font></i>: 
( <font color="Maroon" title="c7">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 ;<br/><a NAME="E4:39_1_5_1_6"/><i><font color="Green" title="E44">A28</font></i>: 
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a1</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b1</font> )
 <b>by </b><i><a class="txt" href="#E3:39_1_5_1_6"><i><font color="Green" title="E43">A27</font></i></a>, <a class="ref" href="topreal6.html#T22" title="TOPREAL6:th.22">TOPREAL6:22</a></i>;<br/><a NAME="E5:39_1_5_1_6"/>
<font color="Maroon" title="c7">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E5:39"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="#E3:39_1_5_1_6"><i><font color="Green" title="E43">A27</font></i></a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></i>;<br/><a NAME="E6:39_1_5_1_6"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="#E4:39_1_5"><i><font color="Green" title="E42">A20</font></i></a>, <a class="txt" href="#E4:39_1_5_1_6"><i><font color="Green" title="E44">A28</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E7:39_1_5_1_6"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E9:39"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="#E3:39_1_5"><i><font color="Green" title="E41">A19</font></i></a>, <a class="txt" href="#E4:39_1_5"><i><font color="Green" title="E42">A20</font></i></a>, <a class="ref" href="xxreal_1.html#T2" title="XXREAL_1:th.2">XXREAL_1:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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="39_1_6"><b>suppose </b></a><a NAME="E1:39_1_6"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:39_1_6"/><i><font color="Green" title="E41">A29</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 ;<br/><a NAME="E4:39_1_6"/><i><font color="Green" title="E42">A30</font></i>: 
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="txt" href="#E3:39_1_6"><i><font color="Green" title="E41">A29</font></i></a>, <a class="ref" href="topreal6.html#T22" title="TOPREAL6:th.22">TOPREAL6:22</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_6_1"/>
( <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  or  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Olive" title="b1">a</font> or  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Olive" title="b1">a</font> or  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Olive" title="b1">a</font> or  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Olive" title="b1">a</font> or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> ) or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> ) or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> ) or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> ) )
 </span><b>by </b><i><a class="ref" href="rcomp_3.html#T41" target="_self" title="RCOMP_3:th.41">Th41</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_6_1_1"><b>suppose </b></a><a NAME="E1:39_1_6_1_1"/>
<font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><b>hence </b><a NAME="E2:39_1_6_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_6_1_2"><b>suppose </b></a><a NAME="E1:39_1_6_1_2"/>
(  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K2" title="LIMFUNC1:func.2">left_closed_halfline</a> <font color="Olive" title="b1">a</font> or  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#NK2" title="LIMFUNC1:NK.2">left_open_halfline</a> <font color="Olive" title="b1">a</font> or  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Olive" title="b1">a</font> or  ex <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Olive" title="b1">a</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><b>hence </b><a NAME="E2:39_1_6_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_6_1_3"><b>suppose </b></a><a NAME="E1:39_1_6_1_3"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c8">b1</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:39_1_6_1_3"/><i><font color="Green" title="E43">A31</font></i>: 
( <font color="Maroon" title="c7">a1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">b1</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 ;<br/><a NAME="E4:39_1_6_1_3"/><i><font color="Green" title="E44">A32</font></i>: 
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a1</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b1</font> )
 <b>by </b><i><a class="txt" href="#E3:39_1_6_1_3"><i><font color="Green" title="E43">A31</font></i></a>, <a class="ref" href="jordan5a.html#T20" title="JORDAN5A:th.20">JORDAN5A:20</a></i>;<br/><a NAME="E5:39_1_6_1_3"/><i><font color="Green" title="E45">A33</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">b1</font>
 <b>by </b><i><a class="txt" href="#E5:39"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="#E3:39_1_6_1_3"><i><font color="Green" title="E43">A31</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/><a NAME="E6:39_1_6_1_3"/>
<font color="Maroon" title="c8">b1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="#E2:39"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="#E4:39"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="#E3:39_1_6"><i><font color="Green" title="E41">A29</font></i></a>, <a class="txt" href="#E4:39_1_6"><i><font color="Green" title="E42">A30</font></i></a>, <a class="txt" href="#E3:39_1_6_1_3"><i><font color="Green" title="E43">A31</font></i></a>, <a class="txt" href="#E4:39_1_6_1_3"><i><font color="Green" title="E44">A32</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></i>;<br/><a NAME="E7:39_1_6_1_3"/><b>then </b><i><font color="Green" title="E46">A34</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="#E5:39_1_6_1_3"><i><font color="Green" title="E45">A33</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E8:39_1_6_1_3"/>
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">a1</font>
 <b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="#E3:39"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="#E3:39_1_6"><i><font color="Green" title="E41">A29</font></i></a>, <a class="txt" href="#E4:39_1_6"><i><font color="Green" title="E42">A30</font></i></a>, <a class="txt" href="#E3:39_1_6_1_3"><i><font color="Green" title="E43">A31</font></i></a>, <a class="txt" href="#E4:39_1_6_1_3"><i><font color="Green" title="E44">A32</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></i>;<br/><a NAME="E9:39_1_6_1_3"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E8:39"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="#E4:39_1_6_1_3"><i><font color="Green" title="E44">A32</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E10:39_1_6_1_3"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E3:39_1_6"><i><font color="Green" title="E41">A29</font></i></a>, <a class="txt" href="#E7:39_1_6_1_3"><i><font color="Green" title="E46">A34</font></i></a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_6_1_4"><b>suppose </b></a><a NAME="E1:39_1_6_1_4"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c8">b1</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:39_1_6_1_4"/><i><font color="Green" title="E43">A35</font></i>: 
( <font color="Maroon" title="c7">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> )
 ;<br/><a NAME="E4:39_1_6_1_4"/><i><font color="Green" title="E44">A36</font></i>: 
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a1</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b1</font> )
 <b>by </b><i><a class="txt" href="#E3:39_1_6_1_4"><i><font color="Green" title="E43">A35</font></i></a>, <a class="ref" href="rcomp_3.html#T4" target="_self" title="RCOMP_3:th.4">Th4</a>, <a class="ref" href="rcomp_3.html#T5" target="_self" title="RCOMP_3:th.5">Th5</a></i>;<br/><a NAME="E5:39_1_6_1_4"/>
<font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font>
 <b>by </b><i><a class="txt" href="#E5:39"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="#E3:39_1_6_1_4"><i><font color="Green" title="E43">A35</font></i></a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</a></i>;<br/><a NAME="E6:39_1_6_1_4"/><b>then </b><i><font color="Green" title="E45">A37</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="#E2:39"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="#E4:39_1_6"><i><font color="Green" title="E42">A30</font></i></a>, <a class="txt" href="#E4:39_1_6_1_4"><i><font color="Green" title="E44">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:39_1_6_1_4"/><i><font color="Green" title="E46">A38</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">a1</font>
 <b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="#E3:39"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="#E3:39_1_6"><i><font color="Green" title="E41">A29</font></i></a>, <a class="txt" href="#E4:39_1_6"><i><font color="Green" title="E42">A30</font></i></a>, <a class="txt" href="#E3:39_1_6_1_4"><i><font color="Green" title="E43">A35</font></i></a>, <a class="txt" href="#E4:39_1_6_1_4"><i><font color="Green" title="E44">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></i>;<br/><a NAME="E8:39_1_6_1_4"/>
<font color="Maroon" title="c7">a1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E5:39"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="#E3:39_1_6_1_4"><i><font color="Green" title="E43">A35</font></i></a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</a></i>;<br/><a NAME="E9:39_1_6_1_4"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E7:39_1_6_1_4"><i><font color="Green" title="E46">A38</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E10:39_1_6_1_4"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E3:39_1_6"><i><font color="Green" title="E41">A29</font></i></a>, <a class="txt" href="#E6:39_1_6_1_4"><i><font color="Green" title="E45">A37</font></i></a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_6_1_5"><b>suppose </b></a><a NAME="E1:39_1_6_1_5"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c8">b1</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:39_1_6_1_5"/><i><font color="Green" title="E43">A39</font></i>: 
( <font color="Maroon" title="c7">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</a><span class="default"><font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">.]</a></span> )
 ;<br/><a NAME="E4:39_1_6_1_5"/><i><font color="Green" title="E44">A40</font></i>: 
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a1</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b1</font> )
 <b>by </b><i><a class="txt" href="#E3:39_1_6_1_5"><i><font color="Green" title="E43">A39</font></i></a>, <a class="ref" href="rcomp_3.html#T6" target="_self" title="RCOMP_3:th.6">Th6</a>, <a class="ref" href="rcomp_3.html#T7" target="_self" title="RCOMP_3:th.7">Th7</a></i>;<br/><a NAME="E5:39_1_6_1_5"/><i><font color="Green" title="E45">A41</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">b1</font>
 <b>by </b><i><a class="txt" href="#E5:39"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="#E3:39_1_6_1_5"><i><font color="Green" title="E43">A39</font></i></a>, <a class="ref" href="xxreal_1.html#T2" title="XXREAL_1:th.2">XXREAL_1:2</a></i>;<br/><a NAME="E6:39_1_6_1_5"/>
<font color="Maroon" title="c8">b1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="#E2:39"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="#E4:39"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="#E3:39_1_6"><i><font color="Green" title="E41">A29</font></i></a>, <a class="txt" href="#E4:39_1_6"><i><font color="Green" title="E42">A30</font></i></a>, <a class="txt" href="#E3:39_1_6_1_5"><i><font color="Green" title="E43">A39</font></i></a>, <a class="txt" href="#E4:39_1_6_1_5"><i><font color="Green" title="E44">A40</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a>, <a class="ref" href="xxreal_1.html#T2" title="XXREAL_1:th.2">XXREAL_1:2</a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></i>;<br/><a NAME="E7:39_1_6_1_5"/><b>then </b><i><font color="Green" title="E46">A42</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="#E5:39_1_6_1_5"><i><font color="Green" title="E45">A41</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E8:39_1_6_1_5"/>
<font color="Maroon" title="c7">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E5:39"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="#E3:39_1_6_1_5"><i><font color="Green" title="E43">A39</font></i></a>, <a class="ref" href="xxreal_1.html#T2" title="XXREAL_1:th.2">XXREAL_1:2</a></i>;<br/><a NAME="E9:39_1_6_1_5"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="#E4:39_1_6"><i><font color="Green" title="E42">A30</font></i></a>, <a class="txt" href="#E4:39_1_6_1_5"><i><font color="Green" title="E44">A40</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E10:39_1_6_1_5"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E3:39_1_6"><i><font color="Green" title="E41">A29</font></i></a>, <a class="txt" href="#E7:39_1_6_1_5"><i><font color="Green" title="E46">A42</font></i></a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_6_1_6"><b>suppose </b></a><a NAME="E1:39_1_6_1_6"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">a1</font>, <font color="Maroon" title="c8">b1</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:39_1_6_1_6"/><i><font color="Green" title="E43">A43</font></i>: 
( <font color="Maroon" title="c7">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font> &amp; <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 ;<br/><a NAME="E4:39_1_6_1_6"/><i><font color="Green" title="E44">A44</font></i>: 
(  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a1</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b1</font> )
 <b>by </b><i><a class="txt" href="#E3:39_1_6_1_6"><i><font color="Green" title="E43">A43</font></i></a>, <a class="ref" href="topreal6.html#T22" title="TOPREAL6:th.22">TOPREAL6:22</a></i>;<br/><a NAME="E5:39_1_6_1_6"/>
<font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b1</font>
 <b>by </b><i><a class="txt" href="#E5:39"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="#E3:39_1_6_1_6"><i><font color="Green" title="E43">A43</font></i></a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></i>;<br/><a NAME="E6:39_1_6_1_6"/><b>then </b><i><font color="Green" title="E45">A45</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="#E2:39"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="#E4:39_1_6"><i><font color="Green" title="E42">A30</font></i></a>, <a class="txt" href="#E4:39_1_6_1_6"><i><font color="Green" title="E44">A44</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:39_1_6_1_6"/>
<font color="Maroon" title="c7">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E5:39"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="#E3:39_1_6_1_6"><i><font color="Green" title="E43">A43</font></i></a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></i>;<br/><a NAME="E8:39_1_6_1_6"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="#E4:39_1_6"><i><font color="Green" title="E42">A30</font></i></a>, <a class="txt" href="#E4:39_1_6_1_6"><i><font color="Green" title="E44">A44</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E9:39_1_6_1_6"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="#E3:39_1_6"><i><font color="Green" title="E41">A29</font></i></a>, <a class="txt" href="#E6:39_1_6_1_6"><i><font color="Green" title="E45">A45</font></i></a>, <a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
