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<div class="add">

<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">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</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="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="E1:43"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">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>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/> <a href="topreal6.html#K1">dist</a> <font color="Olive">p</font>,<font color="Olive">q</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 ;<br/>

<a NAME="E2:43"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</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>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Olive">p</font>,<font color="Olive">q</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:43_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">a</font>, <font color="Maroon">q</font> = <font color="Maroon">b</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/>
<a NAME="E3:43_1"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</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> st <br/>( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp;  <a href="topreal6.html#K1">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <font color="Olive">a</font>,<font color="Olive">b</font> )
 
<b>by </b><i><a class="ref" href="topreal6.html#D1">TOPREAL6:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:43_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:43"/>
 ex <font color="Olive">A'</font>, <font color="Olive">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> st <br/>( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">A'</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">B'</font> &amp;  <a href="jordan1k.html#K1" 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="Olive">A'</font>,<font color="Olive">B'</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:43"/>
 <a href="jordan1k.html#K1" target="_self">dist_min</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/>


</div>
