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<b>let </b><font color="Maroon" title="c1">n</font> be    <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> ;<br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be  <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:42"/><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">B</font>
 ;<br/>

<a NAME="E2:42"/>
 ex <font color="Olive" title="b1">A'</font>, <font color="Olive" title="b2">B'</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> st <br/>( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A'</font> &amp; <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B'</font> &amp;  <a href="jordan1k.html#K1" title="JORDAN1K:func.1">dist_min</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="weierstr.html#K9" title="WEIERSTR:func.9">min_dist_min</a> <font color="Olive" title="b1">A'</font>,<font color="Olive" title="b2">B'</font> )
 
<b>by </b><i><a class="ref" href="jordan1k.html#D1" target="_self" title="JORDAN1K:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E3:42"/>
 <a href="jordan1k.html#K1" title="JORDAN1K:func.1">dist_min</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="jordan1k.html#E1:42"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="jordan1k.html#T12" target="_self" title="JORDAN1K:th.12">Th12</a></i>;<br/>


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