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<b>let </b><font color="Maroon" title="c1">M</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">Q</font>, <font color="Maroon" title="c3">R</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>;<br/><b>let </b><font color="Maroon" title="c4">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:37"/><i><font color="Green" title="E30">A1</font></i>: 
( <font color="Maroon" title="c2">Q</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> &amp; <font color="Maroon" title="c3">R</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> )
 <b>and </b><br/><a NAME="E2:37"/><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Q</font>
 ;<br/>

<a NAME="E3:37"/><i><font color="Green" title="E32">A3</font></i>: 
 <a href="weierstr.html#K10" title="WEIERSTR:func.10">max_dist_min</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c2">Q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="weierstr.html#K10" title="WEIERSTR:func.10">max_dist_min</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c2">Q</font></span>)</span>,<span class="p1">(<span class="default"><a href="weierstr.html#K10" title="WEIERSTR:func.10">max_dist_min</a> <font color="Maroon" title="c2">Q</font>,<font color="Maroon" title="c3">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/>
<a NAME="E4:37"/>
<span class="p1">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="weierstr.html#K10" title="WEIERSTR:func.10">max_dist_min</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c2">Q</font>
 
<b>by </b><i><a class="txt" href="hausdorf.html#E1:37"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="hausdorf.html#E2:37"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="hausdorf.html#T33" target="_self" title="HAUSDORF:th.33">Th33</a></i>;<br/>
<a NAME="E5:37"/><b>then </b>
<span class="p1">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K2" title="SQUARE_1:func.2">max</a> <span class="p1">(<span class="default"><a href="weierstr.html#K10" title="WEIERSTR:func.10">max_dist_min</a> <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c2">Q</font></span>)</span>,<span class="p1">(<span class="default"><a href="weierstr.html#K10" title="WEIERSTR:func.10">max_dist_min</a> <font color="Maroon" title="c2">Q</font>,<font color="Maroon" title="c3">R</font></span>)</span>
 
<b>by </b><i><a class="txt" href="hausdorf.html#E3:37"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E6:37"/>
<span class="p1">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</a> <font color="Maroon" title="c3">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="hausdorf.html#K1" title="HAUSDORF:func.1">HausDist</a> <font color="Maroon" title="c2">Q</font>,<font color="Maroon" title="c3">R</font>
 ;<br/>


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