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<b>let </b><font color="Maroon">p1</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">p2</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<b>assume </b><a NAME="E1:52"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">p1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font>
 ;<br/>

<a NAME="E2:52"/><i><font color="Green">E32</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">p2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<b>set </b><font color="Maroon">S</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="rcomp_1.html#K1">.]</a></span>;<br/>
<b>reconsider </b><font color="Maroon">S</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="rcomp_1.html#K1">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/>
<a NAME="E4:52"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="borsuk_4.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E5:52"/><i><font color="Green">E47</font></i>: 
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> is <a href="connsp_1.html#V1">connected</a>
 
<b>by </b><i><a class="ref" href="borsuk_4.html#T5" target="_self">Th5</a>, <a class="ref" href="treal_1.html#T23">TREAL_1:23</a></i>;<br/>
<a NAME="E6:52"/>
<a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 
<b>by </b><i><a class="ref" href="borsuk_4.html#T5" target="_self">Th5</a>, , <a class="ref" href="topmetr.html#T31">TOPMETR:31</a></i>;<br/>
<b>hence </b><a NAME="E7:52"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="rcomp_1.html#K1">.]</a></span> is  non <a href="xboole_0.html#V1">empty</a> <a href="connsp_1.html#V2">connected</a> <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> 
 <b>by </b><i><a class="ref" href="borsuk_4.html#T5" target="_self">Th5</a>, , , <a class="ref" href="compts_1.html#T17">COMPTS_1:17</a>, <a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>


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