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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">B</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:12"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">B</font>
 ;<br/>

<b>consider </b><font color="Maroon">A'</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span></span>)</span>, <font color="Maroon">B'</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:12"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B'</font> )
 <b>and </b><br/><a NAME="E4:12"/><i><font color="Green">E44</font></i>: 
 <a href="jordan1k.html#K4">dist_min</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">A'</font>,<font color="Maroon">B'</font>
 <b>by </b><i><a class="ref" href="jordan1k.html#D1">JORDAN1K:def 1</a></i>;<br/>
<b>thus </b><a NAME="E5:12"/>
 <a href="jordan1k.html#K4">dist_min</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, , , <a class="ref" href="jgraph_1.html#T55">JGRAPH_1:55</a></i>;<br/>


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