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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>



<span class="kw">assume </span><a NAME="E1:84"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">thus </span><a NAME="E2:84"/>
( <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="84_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:84_1"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="84_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:84_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="84_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:84_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"><span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E39">A1</font></span></a></span>;<br/><a NAME="E3:84_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/><span class="kw">hence </span><a NAME="E4:84_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:84_1_1_2"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"><span class="kw">reconsider </span><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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</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> <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T17" title="TOPMETR:th.17">TOPMETR:17</a></span>;<br/><a NAME="E3:84_1_1_2"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
<font color="Maroon" title="c1">A</font> is  <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84_1"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="rcomp_1.html#T10" title="RCOMP_1:th.10">RCOMP_1:10</a></span>;<br/><a NAME="E4:84_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A5</font></span>: 
 <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84_1_1_2"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="seq_4.html#T11" title="SEQ_4:th.11">SEQ_4:11</a>, <a class="ref" href="topmetr.html#T19" title="TOPMETR:th.19">TOPMETR:19</a></span>;<br/><a NAME="E5:84_1_1_2"/><span class="lab"><font color="Green" title="E44">A6</font></span>: 
<font color="Maroon" title="c2">B</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E1:84_1"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="topreal6.html#T70" target="_self" title="TOPREAL6:th.70">Th79</a></span>;<br/><a NAME="E6:84_1_1_2"/><span class="lab"><font color="Green" title="E45">A7</font></span>: 
( <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="E7:84_1_1_2"/>
 <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>) is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84_1_1_2"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="txt" href="#E3:84_1_1_2"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="heine.html#T4" title="HEINE:th.4">HEINE:4</a>, <a class="ref" href="seq_4.html#T11" title="SEQ_4:th.11">SEQ_4:11</a></span>;<br/><a NAME="E8:84_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">X</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:84_1_1_2"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="txt" href="#E4:84_1_1_2"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="ref" href="compts_1.html#T3" title="COMPTS_1:th.3">COMPTS_1:3</a></span>;<br/><span class="kw">hence </span><a NAME="E9:84_1_1_2"/>
<font color="Maroon" title="c2">B</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E5:84_1_1_2"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="txt" href="#E3:84_1_1_2"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="topreal6.html#T18" target="_self" title="TOPREAL6:th.18">Th24</a>, <a class="ref" href="compts_1.html#T9" title="COMPTS_1:th.9">COMPTS_1:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E3:84"/>
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E4:84"/><span class="kw">then </span>
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="weierstr.html#T13" title="WEIERSTR:th.13">WEIERSTR:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:84"/>
<font color="Maroon" title="c1">A</font> is  <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="weierstr.html#D1" title="WEIERSTR:def.1">WEIERSTR:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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