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<b>let </b><font color="Maroon" title="c1">M</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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> <font color="Maroon" title="c1">M</font></span>)</span> holds   <a href="weierstr.html#K8" title="WEIERSTR:func.8">max_dist_min</a> <font color="Olive" title="b1">P</font>,<font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><b>let </b><font color="Maroon" title="c2">P</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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> <font color="Maroon" title="c1">M</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="weierstr.html#K8" title="WEIERSTR:func.8">max_dist_min</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>



<i><font color="Green" title="E27">A1</font></i>:  <a href="weierstr.html#K1" title="WEIERSTR:func.1">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">P</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">P</font>
<b>by </b><i><a class="ref" href="weierstr.html#D3" title="WEIERSTR:def.3">WEIERSTR:def 3</a></i>
<br/>.= 
<span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
<b>by </b><i><a class="ref" href="hausdorf.html#T29" target="_self" title="HAUSDORF:th.29">Th29</a></i>
;<br/>
 <a href="weierstr.html#K8" title="WEIERSTR:func.8">max_dist_min</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c2">P</font> = 
 <a href="weierstr.html#K2" title="WEIERSTR:func.2">upper_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">P</font></span>)</span>
<b>by </b><i><a class="ref" href="weierstr.html#D10" title="WEIERSTR:def.10">WEIERSTR:def 10</a></i>
<br/>.= 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
<b>by </b><i><a class="txt" href="#E1:33"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="weierstr.html#D4" title="WEIERSTR:def.4">WEIERSTR:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E3:33"/>
 <a href="weierstr.html#K8" title="WEIERSTR:func.8">max_dist_min</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="seq_4.html#T22" title="SEQ_4:th.22">SEQ_4:22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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