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<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">s</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><a NAME="E1:21"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E2:21"/><b>then </b>
 ex <font color="Olive">B</font> being   <a href="cantor_1.html#M1">Basis</a> of  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> st <br/>(  ex <font color="Olive">f</font> being  <a href="struct_0.html#NM8">ManySortedSet</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> st <br/>for <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> holds <br/> ( <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">y</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Olive">n</font></span>)</span></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 </span>}</span>  &amp; <font color="Olive">B</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K3">Union</a> <font color="Olive">f</font> ) &amp; ( for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>  st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Olive">B</font> holds <br/><font color="Olive">X</font> is <a href="connsp_1.html#V2">connected</a> ) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:21"/>
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> is <a href="connsp_2.html#V1">locally_connected</a>
 <b>by </b><i/>;<br/>


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