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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>;<br/><b>let </b><font color="Maroon">p</font> be   <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>;<br/>





<b>consider </b><font color="Maroon">q</font> being   <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>, <font color="Maroon">p'</font> being   <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> such that </b><br/><a NAME="E2:49"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E3:49"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">p'</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E4:49"/><i><font color="Green">E42</font></i>: 
 <a href="jordan1k.html#K4" target="_self">dist_min</a> <font color="Maroon">A</font>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="topreal6.html#K1">dist</a> <font color="Maroon">q</font>,<font color="Maroon">p'</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">q</font>
;<br/>

<b>thus </b><a NAME="E5:49"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="jordan1k.html#T3" target="_self">Th3</a></i>;<br/>

<b>thus </b><a NAME="E6:49"/>
 <a href="jordan1k.html#K5" target="_self">dist</a> <font color="Maroon">p</font>,<font color="Maroon">A</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="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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