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<b>let </b><font color="Maroon">M</font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#NM1">MetrSpace</a>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:34"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">P</font> is <a href="compts_1.html#V6">compact</a> )
 ;<br/>

<b>let </b><font color="Maroon">p1</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>, <font color="Maroon">p2</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">x1</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">x2</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E2:34"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8" target="_self">dist_min</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8" target="_self">dist_min</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p2</font> )
 ;<br/>

<a NAME="E3:34"/>
( <span class="p1">(<span class="default"><a href="weierstr.html#K8" target="_self">dist_min</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="weierstr.html#K5" target="_self">lower_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6" target="_self">dist</a> <font color="Maroon">p1</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="weierstr.html#K8" target="_self">dist_min</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a>  <a href="weierstr.html#K5" target="_self">lower_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6" target="_self">dist</a> <font color="Maroon">p2</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:34"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i>, , </i>;<br/>


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