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<b>let </b><font color="Maroon" title="c1">A</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>   st <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> holds <br/>( <font color="Maroon" title="c1">A</font> is <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a> iff <font color="Olive" title="b1">B</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> )</span><br/><b>let </b><font color="Maroon" title="c2">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font> implies ( <font color="Maroon" title="c1">A</font> is <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a> iff <font color="Maroon" title="c2">B</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> ) )</span><br/>



<b>assume </b><a NAME="E1:83"/><i><font color="Green" title="E39">A1</font></i>: 
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">A</font> is <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a> iff <font color="Maroon" title="c2">B</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> )</span><br/>

<b>thus </b><a NAME="E2:83"/>
( <font color="Maroon" title="c1">A</font> is <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a> implies <font color="Maroon" title="c2">B</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">B</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> implies <font color="Maroon" title="c1">A</font> is <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a> )</span><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" title="E40">A2</font></i>: 
<font color="Maroon" title="c1">A</font> is <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">B</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a></span><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" title="c1">A</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">A</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="83_1_1_1"><b>suppose </b></a><a NAME="E1:83_1_1_1"/>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">B</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c3">C</font> = <font color="Maroon" title="c2">B</font> as  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="txt" href="#E1:83"><i><font color="Green" title="E39">A1</font></i></a></i>;<br/><a NAME="E3:83_1_1_1"/>
<font color="Maroon" title="c3">C</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 ;<br/><b>hence </b><a NAME="E4:83_1_1_1"/>
<font color="Maroon" title="c2">B</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="E41">A3</font></i>: 
<font color="Maroon" title="c1">A</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">B</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a></span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c3">X</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#NK3" title="PSCOMP_1:NK.3">inf</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#NK2" title="PSCOMP_1:NK.2">sup</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/><a NAME="E3:83_1_1_2"/>
<font color="Maroon" title="c1">A</font> is <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a>
 <b>by </b><i><a class="txt" href="#E1:83_1"><i><font color="Green" title="E40">A2</font></i></a></i>;<br/><a NAME="E4:83_1_1_2"/><b>then </b><i><font color="Green" title="E42">A4</font></i>: 
<font color="Maroon" title="c2">B</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="#E1:83"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="topreal6.html#T79" target="_self" title="TOPREAL6:th.79">Th79</a></i>;<br/><a NAME="E5:83_1_1_2"/><i><font color="Green" title="E43">A5</font></i>: 
<font color="Maroon" title="c1">A</font> is <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a>
 <b>by </b><i><a class="txt" href="#E1:83_1"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="rcomp_1.html#T28" title="RCOMP_1:th.28">RCOMP_1:28</a></i>;<br/><a NAME="E6:83_1_1_2"/><b>then </b><i><font color="Green" title="E44">A6</font></i>: 
 <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#NK3" title="PSCOMP_1:NK.3">inf</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#NK2" title="PSCOMP_1:NK.2">sup</a> <font color="Maroon" title="c1">A</font></span>)</span> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>
 <b>by </b><i><a class="txt" href="#E1:83_1_1_2"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="heine.html#T11" title="HEINE:th.11">HEINE:11</a>, <a class="ref" href="seq_4.html#T24" title="SEQ_4:th.24">SEQ_4:24</a></i>;<br/><a NAME="E7:83_1_1_2"/><i><font color="Green" title="E45">A7</font></i>: 
( <font color="Maroon" title="c3">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="c3">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/><a NAME="E8:83_1_1_2"/>
 <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#NK3" title="PSCOMP_1:NK.3">inf</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#NK2" title="PSCOMP_1:NK.2">sup</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="#E1:83_1_1_2"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="#E5:83_1_1_2"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="seq_4.html#T24" title="SEQ_4:th.24">SEQ_4:24</a>, <a class="ref" href="topmetr.html#T26" title="TOPMETR:th.26">TOPMETR:26</a></i>;<br/><a NAME="E9:83_1_1_2"/><b>then </b>
<font color="Maroon" title="c3">X</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 <b>by </b><i><a class="txt" href="#E6:83_1_1_2"><i><font color="Green" title="E44">A6</font></i></a>, <a class="txt" href="#E7:83_1_1_2"><i><font color="Green" title="E45">A7</font></i></a>, <a class="ref" href="compts_1.html#T12" title="COMPTS_1:th.12">COMPTS_1:12</a></i>;<br/><b>hence </b><a NAME="E10:83_1_1_2"/>
<font color="Maroon" title="c2">B</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 <b>by </b><i><a class="txt" href="#E1:83"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="#E4:83_1_1_2"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="#E5:83_1_1_2"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="topreal6.html#T24" target="_self" title="TOPREAL6:th.24">Th24</a>, <a class="ref" href="compts_1.html#T18" title="COMPTS_1:th.18">COMPTS_1:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="c2">B</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">A</font> is <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a></span><br/>

<a NAME="E4:83"/><b>then </b>
 <a href="weierstr.html#K1" title="WEIERSTR:func.1">[#]</a> <font color="Maroon" title="c2">B</font> is <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a>
 
<b>by </b><i><a class="ref" href="weierstr.html#T19" title="WEIERSTR:th.19">WEIERSTR:19</a></i>;<br/>
<b>hence </b><a NAME="E5:83"/>
<font color="Maroon" title="c1">A</font> is <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a>
 <b>by </b><i><a class="txt" href="#E1:83"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="weierstr.html#D3" title="WEIERSTR:def.3">WEIERSTR:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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