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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>, <font color="Maroon" title="c3">Q</font> be   <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>assume </b><a NAME="E1:43"/><i><font color="Green" title="E37">A1</font></i>: 
( <font color="Maroon" title="c2">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c2">P</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> &amp; <font color="Maroon" title="c3">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">Q</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E3:43"/><i><font color="Green" title="E38">A2</font></i>: 
( <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="weierstr.html#K9" title="WEIERSTR:func.9">min_dist_min</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">Q</font> )
 <b>by </b><i><a class="ref" href="weierstr.html#T36" title="WEIERSTR:th.36">WEIERSTR:36</a></i>;<br/>
<a NAME="E4:43"/><i><font color="Green" title="E39">A3</font></i>: 
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V3" title="METRIC_1:attr.3">discerning</a>
 
<b>by </b><i><a class="ref" href="metric_1.html#D8" title="METRIC_1:def.8">METRIC_1:def 8</a></i>;<br/>
<div><i><font color="Green" title="E40">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_1"/>
not  <a href="weierstr.html#K9" title="WEIERSTR:func.9">min_dist_min</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">Q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><a NAME="E2:43_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="jgraph_1.html#E1:43"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="jgraph_1.html#E3:43"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="jgraph_1.html#T54" target="_self" title="JGRAPH_1:th.54">Th54</a></i>;<br/><a NAME="E3:43_1"/><b>then </b>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/><a NAME="E4:43_1"/><b>then </b>
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font>
 <b>by </b><i><a class="txt" href="jgraph_1.html#E4:43"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="metric_1.html#D4" title="METRIC_1:def.4">METRIC_1:def 4</a></i>;<br/><b>hence </b><a NAME="E5:43_1"/>
<font color="Maroon" title="c2">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="jgraph_1.html#E3:43"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_2"/><i><font color="Green" title="E41">A5</font></i>: 
<font color="Maroon" title="c2">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon" title="c6">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">Q</font>;<br/><a NAME="E3:43_2"/><i><font color="Green" title="E42">A6</font></i>: 
( <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E1:43_2"><i><font color="Green" title="E41">A5</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">p'</font> = <font color="Maroon" title="c6">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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>reconsider </b><font color="Maroon" title="c8">q</font> = <font color="Maroon" title="c7">p'</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> <b>by </b><i><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></i>;<br/><a NAME="E6:43_2"/>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V2" title="METRIC_1:attr.2">Reflexive</a>
 <b>by </b><i><a class="ref" href="metric_1.html#D7" title="METRIC_1:def.7">METRIC_1:def 7</a></i>;<br/><a NAME="E7:43_2"/><b>then </b>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="metric_1.html#D3" title="METRIC_1:def.3">METRIC_1:def 3</a></i>;<br/><a NAME="E8:43_2"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/><b>hence </b><a NAME="E9:43_2"/>
not  <a href="weierstr.html#K9" title="WEIERSTR:func.9">min_dist_min</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">Q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jgraph_1.html#E1:43"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="jgraph_1.html#E3:43_2"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="weierstr.html#T40" title="WEIERSTR:th.40">WEIERSTR:40</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:43"/>
( <font color="Maroon" title="c2">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">Q</font> iff  <a href="weierstr.html#K9" title="WEIERSTR:func.9">min_dist_min</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">Q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E5:43"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>


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