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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:18"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V3" target="_self">bounded</a>
 <b>and </b><br/><a NAME="E2:18"/><i><font color="Green">E22</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:18"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="seq_4.html#E2:18"><i><font color="Green">E22</font></i></a>, <a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<a NAME="E5:18"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V2" target="_self">bounded_below</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V1" target="_self">bounded_above</a> )
 
<b>by </b><i><a class="txt" href="seq_4.html#E1:18"><i><font color="Green">E21</font></i></a>, <a class="ref" href="seq_4.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E6:18"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="seq_4.html#K5" target="_self">lower_bound</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="seq_4.html#E4:18"><i><font color="Green">E23</font></i></a>, <a class="ref" href="seq_4.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E7:18"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_4.html#K4" target="_self">upper_bound</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="seq_4.html#E4:18"><i><font color="Green">E23</font></i></a>, <a class="txt" href="seq_4.html#E5:18"><i><font color="Green">E24</font></i></a>, <a class="ref" href="seq_4.html#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E8:18"/>
 <a href="seq_4.html#K5" target="_self">lower_bound</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_4.html#K4" target="_self">upper_bound</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="seq_4.html#E6:18"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div>
