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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:14"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:14"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="weierstr.html#K5">lower_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#NK3" target="_self">inf</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="hausdorf.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E4:14"/><i><font color="Green">E17</font></i>: 
<span class="p1">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#NK3" target="_self">inf</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="hausdorf.html#E3:14"><i><font color="Green">E16</font></i></a>, <a class="ref" href="weierstr.html#D8">WEIERSTR:def 8</a></i>;<br/>
<a NAME="E5:14"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="seq_4.html#V2">bounded_below</a>
 
<b>by </b><i><a class="txt" href="hausdorf.html#E3:14"><i><font color="Green">E16</font></i></a>, <a class="ref" href="hausdorf.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E6:14"/><i><font color="Green">E19</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:14"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="weierstr.html#D6">WEIERSTR:def 6</a></i>;<br/>
<a NAME="E8:14"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="hausdorf.html#E1:14"><i><font color="Green">E15</font></i></a>, <a class="txt" href="hausdorf.html#E3:14"><i><font color="Green">E16</font></i></a>, <a class="txt" href="hausdorf.html#E6:14"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E9:14"/>
<span class="p1">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="hausdorf.html#E4:14"><i><font color="Green">E17</font></i></a>, <a class="txt" href="hausdorf.html#E5:14"><i><font color="Green">E18</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>


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