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

<b>let </b><font color="Maroon" title="c1">X</font> be   <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> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:40"/><i><font color="Green" title="E21">A1</font></i>: 
( <font color="Maroon" title="c1">X</font> is <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> &amp; <font color="Maroon" title="c1">X</font> is <a href="seq_4.html#V3" title="SEQ_4:attr.3">bounded</a> )
 <b>and </b><br/><a NAME="E2:40"/><i><font color="Green" title="E22">A2</font></i>: 
for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</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="Olive" title="b1">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">g1</font>,<font color="Olive" title="b2">g2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1"/>
( <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1"><b>suppose </b></a><a NAME="E1:40_1_1"/><i><font color="Green" title="E23">A3</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>take </b>
1
;<br/><b>take </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
;<br/><b>thus </b><a NAME="E2:40_1_1"/>
<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">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="rcomp_1.html#E1:40_1_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="xxreal_1.html#T28" title="XXREAL_1:th.28">XXREAL_1:28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2"><b>suppose </b></a><a NAME="E1:40_1_2"/><i><font color="Green" title="E23">A4</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:40_1_2"/><i><font color="Green" title="E24">A5</font></i>: 
( <font color="Maroon" title="c1">X</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a> &amp; <font color="Maroon" title="c1">X</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a> )
 <b>by </b><i><a class="txt" href="rcomp_1.html#E1:40"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="seq_4.html#D3" title="SEQ_4:def.3">SEQ_4:def 3</a></i>;<br/><b>take </b><font color="Maroon" title="c2">p</font> =  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">X</font>;<br/><b>take </b><font color="Maroon" title="c3">g</font> =  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">X</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>thus </b><a NAME="E1:40_1_2_1"/>
( <font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">g</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_1_2_1_1"/><i><font color="Green" title="E25">A6</font></i>: 
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<a NAME="E2:40_1_2_1_1"/><b>then </b><i><font color="Green" title="E26">A7</font></i>: 
( <font color="Maroon" title="c3">g</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">r</font> &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">r</font> )
 
<b>by </b><i><a class="txt" href="rcomp_1.html#E1:40"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="rcomp_1.html#E2:40_1_2"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="rcomp_1.html#T43" target="_self" title="RCOMP_1:th.43">Th43</a>, <a class="ref" href="rcomp_1.html#T44" target="_self" title="RCOMP_1:th.44">Th44</a></i>;<br/>
<a NAME="E3:40_1_2_1_1"/>
<font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">g</font>
 
<b>by </b><i><a class="txt" href="rcomp_1.html#E2:40_1_2"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="rcomp_1.html#E1:40_1_2_1_1"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/>
<a NAME="E4:40_1_2_1_1"/><b>then </b><i><font color="Green" title="E27">A8</font></i>: 
<font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">g</font>
 
<b>by </b><i><a class="txt" href="rcomp_1.html#E2:40_1_2_1_1"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E5:40_1_2_1_1"/>
<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="txt" href="rcomp_1.html#E2:40_1_2"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="rcomp_1.html#E1:40_1_2_1_1"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/>
<a NAME="E6:40_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="txt" href="rcomp_1.html#E2:40_1_2_1_1"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:40_1_2_1_1"/>
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">g</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="rcomp_1.html#E4:40_1_2_1_1"><i><font color="Green" title="E27">A8</font></i></a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:40_1_2_1"/>
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">g</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ;<br/><a NAME="E3:40_1_2_1"/><b>then </b><i><font color="Green" title="E25">A9</font></i>: 
 ex <font color="Olive" title="b1">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> &amp; <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">g</font> )
 ;<br/><a NAME="E4:40_1_2_1"/><b>then </b>
<font color="Maroon" title="c3">g</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">g2</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="E6:40_1_2_1"/><i><font color="Green" title="E26">A10</font></i>: 
( <font color="Maroon" title="c5">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c3">g</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">r</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">g2</font> )
 <b>by </b><i><a class="txt" href="rcomp_1.html#E1:40_1_2"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="rcomp_1.html#E2:40_1_2"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/><a NAME="E7:40_1_2_1"/>
<font color="Maroon" title="c4">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="rcomp_1.html#E3:40_1_2_1"><i><font color="Green" title="E25">A9</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">g1</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="E9:40_1_2_1"/><i><font color="Green" title="E27">A11</font></i>: 
( <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">p</font></span>)</span> )
 <b>by </b><i><a class="txt" href="rcomp_1.html#E1:40_1_2"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="rcomp_1.html#E2:40_1_2"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">g1</font> = <font color="Maroon" title="c6">g1</font>, <font color="Maroon" title="c8">g2</font> = <font color="Maroon" title="c5">g2</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><a NAME="E11:40_1_2_1"/><i><font color="Green" title="E28">A12</font></i>: 
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c7">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">g2</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="rcomp_1.html#E6:40_1_2_1"><i><font color="Green" title="E26">A10</font></i></a>, <a class="txt" href="rcomp_1.html#E9:40_1_2_1"><i><font color="Green" title="E27">A11</font></i></a></i>;<br/><a NAME="E12:40_1_2_1"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c7">g1</font>,<font color="Maroon" title="c8">g2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="rcomp_1.html#E2:40"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="rcomp_1.html#E6:40_1_2_1"><i><font color="Green" title="E26">A10</font></i></a>, <a class="txt" href="rcomp_1.html#E9:40_1_2_1"><i><font color="Green" title="E27">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E13:40_1_2_1"/>
<font color="Maroon" title="c4">r</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="rcomp_1.html#E11:40_1_2_1"><i><font color="Green" title="E28">A12</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:40_1_2"/>
<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="c2">p</font>,<font color="Maroon" title="c3">g</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="ref" href="subset_1.html#T8" title="SUBSET_1:th.8">SUBSET_1:8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
