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<span class="kw">let </span><font color="Maroon" title="c1">C</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>   <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> is  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a>  <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> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c2">C9</font> = <font color="Maroon" title="c1">C</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></span>;<br/>
<a NAME="E2:137"/>
<font color="Maroon" title="c1">C</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="ref" href="compts_1.html#T16" title="COMPTS_1:th.16">COMPTS_1:16</a></span>;<br/>
<a NAME="E3:137"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A1</font></span>: 
<font color="Maroon" title="c2">C9</font> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal6.html#T79" title="TOPREAL6:th.79">TOPREAL6:79</a></span>;<br/>
<a NAME="E4:137"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A2</font></span>: 
 <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <font color="Maroon" title="c2">C9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_5.html#T107" target="_self" title="BORSUK_5:th.107">Th107</a>, <a class="ref" href="rcomp_1.html#T30" title="RCOMP_1:th.30">RCOMP_1:30</a></span>;<br/>
<a NAME="E5:137"/>
( <font color="Maroon" title="c2">C9</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a>  &amp; <font color="Maroon" title="c2">C9</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_5.html#T107" target="_self" title="BORSUK_5:th.107">Th107</a></span>;<br/>
<a NAME="E6:137"/><span class="kw">then </span>
<font color="Maroon" title="c2">C9</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="ref" href="xxreal_2.html#D11" title="XXREAL_2:def.11">XXREAL_2:def 11</a></span>;<br/>
<a NAME="E7:137"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A3</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">C9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">C9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T24" title="SEQ_4:th.24">SEQ_4:24</a></span>;<br/>
<a NAME="E8:137"/>
 <a href="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c2">C9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:137"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="ref" href="borsuk_5.html#T107" target="_self" title="BORSUK_5:th.107">Th107</a>, <a class="ref" href="rcomp_1.html#T31" title="RCOMP_1:th.31">RCOMP_1:31</a></span>;<br/>
<a NAME="E9:137"/><span class="kw">then </span>
<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="measure6.html#NK5" title="MEASURE6:NK.5">inf</a> <font color="Maroon" title="c2">C9</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <font color="Maroon" title="c2">C9</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">C9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:137"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="ref" href="borsuk_5.html#T109" target="_self" title="BORSUK_5:th.109">Th109</a>, <a class="ref" href="borsuk_5.html#T111" target="_self" title="BORSUK_5:th.111">Th111</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:137"/>
<font color="Maroon" title="c1">C</font> is  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a>  <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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:137"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="ref" href="integra1.html#D1" title="INTEGRA1:def.1">INTEGRA1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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