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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">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> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:9_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">P</font>
 ;<br/><a NAME="E2:9_1"/><b>then </b>
for <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 holds <br/> ex <font color="Olive" title="b2">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> st <br/>( <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">a</font> )
 <b>by </b><i><a class="ref" href="hausdorf.html#T8" target="_self" title="HAUSDORF:th.8">Th8</a></i>;<br/><b>hence </b><a NAME="E3:9_1"/>
<span class="p1">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</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> 0
 <b>by </b><i><a class="ref" href="hausdorf.html#T9" target="_self" title="HAUSDORF:th.9">Th9</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:9"/>
<span class="p1">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</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> 0
 ;<br/>

<a NAME="E3:9"/><b>then </b>
for <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 holds <br/> ex <font color="Olive" title="b2">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> st <br/>( <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">a</font> )
 
<b>by </b><i><a class="ref" href="hausdorf.html#T9" target="_self" title="HAUSDORF:th.9">Th9</a></i>;<br/>
<b>hence </b><a NAME="E4:9"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">P</font>
 <b>by </b><i><a class="ref" href="hausdorf.html#T8" target="_self" title="HAUSDORF:th.8">Th8</a></i>;<br/>


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