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<b>let </b><font color="Maroon">C</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="connsp_1.html#V2">connected</a> <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> ;<br/>

<b>reconsider </b><font color="Maroon">C'</font> = <font color="Maroon">C</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E2:118"/>
<font color="Maroon">C</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="compts_1.html#T16">COMPTS_1:16</a></i>;<br/>
<a NAME="E3:118"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">C'</font> is <a href="rcomp_1.html#V2">closed</a>
 
<b>by </b><i><a class="ref" href="topreal6.html#T79">TOPREAL6:79</a></i>;<br/>
<a NAME="E4:118"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">C'</font> is <a href="seq_4.html#V2">bounded_below</a> &amp; <font color="Maroon">C'</font> is <a href="seq_4.html#V1">bounded_above</a> )
 
<b>by </b><i><a class="ref" href="borsuk_5.html#T35" target="_self">Th35</a></i>;<br/>
<a NAME="E5:118"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">C'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C'</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T31">RCOMP_1:31</a></i>;<br/>
<a NAME="E6:118"/>
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">C'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C'</font>
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#T30">RCOMP_1:30</a></i>;<br/>
<a NAME="E7:118"/><b>then </b>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">C'</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">C'</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">C'</font>
 
<b>by </b><i>, <a class="ref" href="borsuk_5.html#T37" target="_self">Th37</a></i>;<br/>
<a NAME="E8:118"/><b>then </b><i><font color="Green">E49</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">C'</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">C'</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">C'</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:118"/>
<font color="Maroon">C'</font> is <a href="seq_4.html#V3">bounded</a>
 
<b>by </b><i>, <a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/>
<a NAME="E10:118"/><b>then </b>
 <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">C'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">C'</font>
 
<b>by </b><i><a class="ref" href="seq_4.html#T24">SEQ_4:24</a></i>;<br/>
<b>hence </b><a NAME="E11:118"/>
<font color="Maroon">C</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="integra1.html#V1">closed-interval</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i>, <a class="ref" href="integra1.html#D1">INTEGRA1:def 1</a></i>;<br/>


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