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<b>let </b><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">B</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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>consider </b><font color="Maroon" title="c6">A'</font>, <font color="Maroon" title="c7">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><b> such that </b><br/><a NAME="E2:40_1_1"/><i><font color="Green" title="E30">A1</font></i>: 
( <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A'</font> &amp; <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B'</font> )
 <b>and </b><br/><a NAME="E3:40_1_1"/><i><font color="Green" title="E31">A2</font></i>: 
 <a href="jordan1k.html#K1" title="JORDAN1K:func.1">dist_min</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">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="Maroon" title="c6">A'</font>,<font color="Maroon" title="c7">B'</font>
 <b>by </b><i><a class="ref" href="jordan1k.html#D1" target="_self" title="JORDAN1K:def.1">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">A'</font> = <font color="Maroon" title="c6">A'</font>, <font color="Maroon" title="c9">B'</font> = <font color="Maroon" title="c7">B'</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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="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> <b>by </b><i><a class="txt" href="jordan1k.html#E2:40_1_1"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/>
<a NAME="E5:40_1_1"/>
 <a href="jordan1k.html#K1" title="JORDAN1K:func.1">dist_min</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan1k.html#K2" title="JORDAN1K:func.2">min_dist_min</a> <font color="Maroon" title="c9">B'</font>,<font color="Maroon" title="c8">A'</font>
 
<b>by </b><i><a class="txt" href="jordan1k.html#E3:40_1_1"><i><font color="Green" title="E31">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:40_1_1"/>
 <a href="jordan1k.html#K1" title="JORDAN1K:func.1">dist_min</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan1k.html#K1" title="JORDAN1K:func.1">dist_min</a> <font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="jordan1k.html#E2:40_1_1"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="jordan1k.html#D1" target="_self" title="JORDAN1K:def.1">Def1</a></i>;<br/>


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