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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>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> ;<br/>



<b>assume </b><a NAME="E1:83"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>thus </b><a NAME="E2:83"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="rcomp_1.html#V1">compact</a> implies <font color="Maroon">c<sub>2</sub></font> is <a href="compts_1.html#V6">compact</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:83_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="rcomp_1.html#V1">compact</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_1"><b>suppose </b></a><a NAME="E1:83_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="topreal6.html#E1:83"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E3:83_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="compts_1.html#V6">compact</a>
 ;<br/><b>hence </b><a NAME="E4:83_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="compts_1.html#V6">compact</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_1_1_2"><b>suppose </b></a><a NAME="E1:83_1_1_2"/><i><font color="Green">E45</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> 
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <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<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><a NAME="E3:83_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="rcomp_1.html#V2">closed</a>
 <b>by </b><i><a class="txt" href="topreal6.html#E1:83_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="rcomp_1.html#T26">RCOMP_1:26</a></i>;<br/><a NAME="E4:83_1_1_2"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="topreal6.html#E1:83"><i><font color="Green">E43</font></i></a>, <a class="ref" href="topreal6.html#T79" target="_self">Th79</a></i>;<br/><a NAME="E5:83_1_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V3">bounded</a>
 <b>by </b><i><a class="txt" href="topreal6.html#E1:83_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="rcomp_1.html#T28">RCOMP_1:28</a></i>;<br/><a NAME="E6:83_1_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <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<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="topreal6.html#T24" target="_self">Th24</a></i>;<br/><a NAME="E7:83_1_1_2"/><i><font color="Green">E49</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="topreal6.html#E1:83_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="topreal6.html#E5:83_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="seq_4.html#T24">SEQ_4:24</a></i>;<br/><a NAME="E8:83_1_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="compts_1.html#V2">compact</a>
 <b>by </b><i><a class="ref" href="heine.html#T11">HEINE:11</a></i>;<br/><a NAME="E9:83_1_1_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><a NAME="E10:83_1_1_2"/>
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <a href="topmetr.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="topreal6.html#E7:83_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="topmetr.html#T26">TOPMETR:26</a></i>;<br/><a NAME="E11:83_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="compts_1.html#V6">compact</a>
 <b>by </b><i><a class="txt" href="topreal6.html#E8:83_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="topreal6.html#E9:83_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="compts_1.html#T12">COMPTS_1:12</a></i>;<br/><b>hence </b><a NAME="E12:83_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="compts_1.html#V6">compact</a>
 <b>by </b><i><a class="txt" href="topreal6.html#E1:83"><i><font color="Green">E43</font></i></a>, <a class="txt" href="topreal6.html#E4:83_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="topreal6.html#E6:83_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="compts_1.html#T18">COMPTS_1:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>assume </b><a NAME="E3:83"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="compts_1.html#V6">compact</a>
 ;<br/>

<a NAME="E4:83"/><b>then </b>
 <a href="weierstr.html#K3">[#]</a> <font color="Maroon">c<sub>2</sub></font> is <a href="rcomp_1.html#V1">compact</a>
 
<b>by </b><i><a class="ref" href="weierstr.html#T19">WEIERSTR:19</a></i>;<br/>
<b>hence </b><a NAME="E5:83"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="rcomp_1.html#V1">compact</a>
 <b>by </b><i><a class="txt" href="topreal6.html#E1:83"><i><font color="Green">E43</font></i></a>, <a class="ref" href="weierstr.html#D3">WEIERSTR:def 3</a></i>;<br/>


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