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<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="E1:33"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:33"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="rcomp_1.html#V1" target="_self">compact</a>
 ;<br/>

<a NAME="E3:33"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V3">bounded</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="rcomp_1.html#V2" target="_self">closed</a> )
 
<b>by </b><i><a class="txt" href="rcomp_1.html#E2:33"><i><font color="Green">E19</font></i></a>, <a class="ref" href="rcomp_1.html#T26" target="_self">Th26</a>, <a class="ref" href="rcomp_1.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E4:33"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V1">bounded_above</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V2">bounded_below</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="rcomp_1.html#V2" target="_self">closed</a> )
 
<b>by </b><i><a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:33"/>
(  <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>1</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="rcomp_1.html#E1:33"><i><font color="Green">E18</font></i></a>, <a class="ref" href="rcomp_1.html#T30" target="_self">Th30</a>, <a class="ref" href="rcomp_1.html#T31" target="_self">Th31</a></i>;<br/>


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