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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="firebrick">::  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="measure5.html#V2" title="MEASURE5:attr.2">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#T17" title="TOPMETR:th.17">TOPMETR:17</a></span>;<br/>
<a NAME="E2:98"/>
<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#T7" title="COMPTS_1:th.7">COMPTS_1:7</a></span>;<br/>
<a NAME="E3:98"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">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#T70" title="TOPREAL6:th.70">TOPREAL6:70</a></span>;<br/>
<a NAME="E4:98"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A2</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</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#T73" target="_self" title="BORSUK_5:th.73">Th107</a>, <a class="ref" href="rcomp_1.html#T12" title="RCOMP_1:th.12">RCOMP_1:12</a></span>;<br/>
<a NAME="E5:98"/>
( <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#T73" target="_self" title="BORSUK_5:th.73">Th107</a></span>;<br/>
<a NAME="E6:98"/><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="E7:98"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">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#T11" title="SEQ_4:th.11">SEQ_4:11</a></span>;<br/>
<a NAME="E8:98"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</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="#E3:98"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="ref" href="borsuk_5.html#T73" target="_self" title="BORSUK_5:th.73">Th107</a>, <a class="ref" href="rcomp_1.html#T13" title="RCOMP_1:th.13">RCOMP_1:13</a></span>;<br/>
<a NAME="E9:98"/><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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">C9</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="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="#E4:98"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="ref" href="borsuk_5.html#T75" target="_self" title="BORSUK_5:th.75">Th109</a>, <a class="ref" href="borsuk_5.html#T77" target="_self" title="BORSUK_5:th.77">Th111</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:98"/>
<font color="Maroon" title="c1">C</font> is   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">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="#E7:98"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="ref" href="measure5.html#T14" title="MEASURE5:th.14">MEASURE5:14</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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