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<span class="kw">let </span><font color="Maroon" title="c6">r</font>, <font color="Maroon" title="c7">s</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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="c6">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">s</font> implies  <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c7">s</font>) is  <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:20"/>
<font color="Maroon" title="c6">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c7">s</font>) is  <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a> </span><br/>

<a NAME="E2:20"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">B</font> being   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c7">s</font>)</span>)</span> st <br/>(  ex <font color="Olive" title="b2">f</font> being   <a href="struct_0.html#NM8" title="STRUCT_0:NM.8">ManySortedSet</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c7">s</font>)</span>)</span> st <br/> for <font color="Olive" title="b3">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2" title="TOPMETR:func.2">Closed-Interval-MSpace</a> (<font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c7">s</font>)</span>)</span> holds <br/> ( <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Olive" title="b3">y</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Olive" title="b4">n</font></span>)</span>)</span>)</span> where <font color="Olive" title="b4">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b4">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span> } </span>  &amp; <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Olive" title="b2">f</font> ) &amp; (  for <font color="Olive" title="b2">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c7">s</font>)</span>)</span>  st <font color="Olive" title="b2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Olive" title="b2">X</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_5.html#T2" target="_self" title="TOPALG_5:th.2">Th2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:20"/>
 <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c7">s</font>) is  <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_5.html#T16" target="_self" title="TOPALG_5:th.16">Th16</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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