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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">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>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="E2:45"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font>
 <b>and </b><br/><a NAME="E3:45"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B'</font>
 <b>and </b><br/><a NAME="E4:45"/><i><font color="Green">E42</font></i>: 
 <a href="jordan1k.html#K4" target="_self">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/>;<br/>
<b>consider </b><font color="Maroon">x1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">x2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E6:45"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font>
 <b>and </b><br/><a NAME="E7:45"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font>
 <b>and </b><br/><a NAME="E8:45"/><i><font color="Green">E48</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</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="txt" href="jordan1k.html#E2:45"><i><font color="Green">E36</font></i></a>, <a class="ref" href="weierstr.html#T36">WEIERSTR:36</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x1</font>, <font color="Maroon">q</font> = <font color="Maroon">x2</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<b>take </b>
<font color="Maroon">p</font>
;<br/>

<b>take </b>
<font color="Maroon">q</font>
;<br/>

<b>thus </b><a NAME="E10:45"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i>, <a class="txt" href="jordan1k.html#E2:45"><i><font color="Green">E36</font></i></a>, , </i>;<br/>

<b>thus </b><a NAME="E11:45"/>
 <a href="jordan1k.html#K4" target="_self">dist_min</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="jordan1k.html#T3" target="_self">Th3</a>, <a class="txt" href="jordan1k.html#E3:45"><i><font color="Green">E40</font></i></a>, <a class="ref" href="topreal6.html#D1">TOPREAL6:def 1</a></i>;<br/>


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