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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  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="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>, <font color="Maroon">Q</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="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>, <font color="Maroon">R</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="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:42"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">P</font> is <a href="compts_1.html#V6">compact</a>
 <b>and </b><br/><a NAME="E2:42"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">Q</font> is <a href="compts_1.html#V6">compact</a>
 <b>and </b><br/><a NAME="E3:42"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">R</font> is <a href="compts_1.html#V6">compact</a>
 ;<br/>

<a NAME="E4:42"/><i><font color="Green">E39</font></i>: 
 <a href="weierstr.html#K10">max_dist_min</a> <font color="Maroon">P</font>,<font color="Maroon">R</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" target="_self">HausDist</a> <font color="Maroon">P</font>,<font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" target="_self">HausDist</a> <font color="Maroon">Q</font>,<font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="txt" href="hausdorf.html#E3:42"><i><font color="Green">E36</font></i></a>, , , </i>;<br/>
<a NAME="E5:42"/>
 <a href="weierstr.html#K10">max_dist_min</a> <font color="Maroon">R</font>,<font color="Maroon">P</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" target="_self">HausDist</a> <font color="Maroon">P</font>,<font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" target="_self">HausDist</a> <font color="Maroon">Q</font>,<font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="txt" href="hausdorf.html#E3:42"><i><font color="Green">E36</font></i></a>, , , <a class="ref" href="hausdorf.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E6:42"/><b>then </b>
 <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><a href="weierstr.html#K10">max_dist_min</a> <font color="Maroon">P</font>,<font color="Maroon">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="weierstr.html#K10">max_dist_min</a> <font color="Maroon">R</font>,<font color="Maroon">P</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" target="_self">HausDist</a> <font color="Maroon">P</font>,<font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" target="_self">HausDist</a> <font color="Maroon">Q</font>,<font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="txt" href="hausdorf.html#E4:42"><i><font color="Green">E39</font></i></a>, <a class="ref" href="square_1.html#T50">SQUARE_1:50</a></i>;<br/>
<b>hence </b><a NAME="E7:42"/>
 <a href="hausdorf.html#K1" target="_self">HausDist</a> <font color="Maroon">P</font>,<font color="Maroon">R</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" target="_self">HausDist</a> <font color="Maroon">P</font>,<font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" target="_self">HausDist</a> <font color="Maroon">Q</font>,<font color="Maroon">R</font></span>)</span>
 ;<br/>


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