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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>;<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>;<br/><b>let </b><font color="Maroon" title="c3">x</font>, <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="E1:24"/><i><font color="Green" title="E21">A1</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">P</font>
 <b>and </b><br/><a NAME="E2:24"/><i><font color="Green" title="E22">A2</font></i>: 
<font color="Maroon" title="c2">P</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">X</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E4:24"/><i><font color="Green" title="E23">A3</font></i>: 
( <font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">P</font> &amp;  <a href="weierstr.html#K4" title="WEIERSTR:func.4">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</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#NK2" title="PSCOMP_1:NK.2">sup</a> <font color="Maroon" title="c5">X</font> )
 <b>by </b><i><a class="ref" href="hausdorf.html#T13" target="_self" title="HAUSDORF:th.13">Th13</a></i>;<br/>
<a NAME="E5:24"/><i><font color="Green" title="E24">A4</font></i>: 
<span class="p1">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">dist_max</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#NK2" title="PSCOMP_1:NK.2">sup</a> <font color="Maroon" title="c5">X</font>
 
<b>by </b><i><a class="txt" href="hausdorf.html#E4:24"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="weierstr.html#D7" title="WEIERSTR:def.7">WEIERSTR:def 7</a></i>;<br/>
<a NAME="E6:24"/><i><font color="Green" title="E25">A5</font></i>: 
<font color="Maroon" title="c5">X</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 
<b>by </b><i><a class="txt" href="hausdorf.html#E2:24"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="hausdorf.html#E4:24"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="hausdorf.html#E22"><i><font color="Green" title="E19">Lm1</font></i></a></i>;<br/>
<a NAME="E7:24"/><i><font color="Green" title="E26">A6</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:24"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font>
 
<b>by </b><i><a class="ref" href="weierstr.html#D6" title="WEIERSTR:def.6">WEIERSTR:def 6</a></i>;<br/>
<a NAME="E9:24"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 
<b>by </b><i><a class="txt" href="hausdorf.html#E1:24"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="hausdorf.html#E4:24"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="hausdorf.html#E7:24"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E10:24"/>
<span class="p1">(<span class="default"><a href="weierstr.html#K7" title="WEIERSTR:func.7">dist_max</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="hausdorf.html#E5:24"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="hausdorf.html#E6:24"><i><font color="Green" title="E25">A5</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/>


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