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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   <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="E1:31"/><i><font color="Green" title="E23">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> )
 ;<br/>

<b>let </b><font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">p2</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>consider </b><font color="Maroon" title="c5">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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> such that </b><br/><a NAME="E3:31"/><i><font color="Green" title="E24">A2</font></i>: 
( <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <span class="p1">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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">p1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">P</font></span>)</span> )
 <b>by </b><i><a class="txt" href="weierstr.html#E1:31"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="weierstr.html#T23" target="_self" title="WEIERSTR:th.23">Th23</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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> such that </b><br/><a NAME="E5:31"/><i><font color="Green" title="E25">A3</font></i>: 
( <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <span class="p1">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c4">p2</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">P</font></span>)</span> )
 <b>by </b><i><a class="txt" href="weierstr.html#E1:31"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="weierstr.html#T23" target="_self" title="WEIERSTR:th.23">Th23</a></i>;<br/>
<a NAME="E6:31"/><i><font color="Green" title="E26">A4</font></i>: 
 <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <font color="Maroon" title="c1">M</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="pcomps_1.html#D6" title="PCOMPS_1:def.6">PCOMPS_1:def 6</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">x1</font> = <font color="Maroon" title="c5">x1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">x2</font> = <font color="Maroon" title="c6">x2</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="txt" href="weierstr.html#E6:31"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/>
<a NAME="E9:31"/><i><font color="Green" title="E27">A5</font></i>: 
<span class="p1">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c3">p1</font>
 
<b>by </b><i><a class="ref" href="weierstr.html#D6" target="_self" title="WEIERSTR:def.6">Def6</a></i>;<br/>
<a NAME="E10:31"/><i><font color="Green" title="E28">A6</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c3">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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">p1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">P</font></span>)</span>
 
<b>by </b><i><a class="txt" href="weierstr.html#E3:31"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="weierstr.html#D6" target="_self" title="WEIERSTR:def.6">Def6</a></i>;<br/>
<a NAME="E11:31"/><i><font color="Green" title="E29">A7</font></i>: 
<span class="p1">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c4">p2</font>
 
<b>by </b><i><a class="ref" href="weierstr.html#D6" target="_self" title="WEIERSTR:def.6">Def6</a></i>;<br/>
<a NAME="E12:31"/><i><font color="Green" title="E30">A8</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c4">p2</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">P</font></span>)</span>
 
<b>by </b><i><a class="txt" href="weierstr.html#E5:31"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="weierstr.html#D6" target="_self" title="WEIERSTR:def.6">Def6</a></i>;<br/>
<a NAME="E13:31"/><i><font color="Green" title="E31">A9</font></i>: 
(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c3">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c4">p2</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font></span>)</span> &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c4">p2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c3">p1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a></i>;<br/>
<a NAME="E14:31"/><i><font color="Green" title="E32">A10</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c4">p2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c4">p2</font>
 
<b>by </b><i><a class="txt" href="weierstr.html#E1:31"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="weierstr.html#E3:31"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="weierstr.html#E5:31"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="weierstr.html#E11:31"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="weierstr.html#T25" target="_self" title="WEIERSTR:th.25">Th25</a></i>;<br/>
<a NAME="E15:31"/><i><font color="Green" title="E33">A11</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c3">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c3">p1</font>
 
<b>by </b><i><a class="txt" href="weierstr.html#E1:31"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="weierstr.html#E3:31"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="weierstr.html#E5:31"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="weierstr.html#E9:31"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="weierstr.html#T25" target="_self" title="WEIERSTR:th.25">Th25</a></i>;<br/>
<a NAME="E16:31"/><i><font color="Green" title="E34">A12</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c3">p1</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c4">p2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c4">p2</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c4">p2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="weierstr.html#E13:31"><i><font color="Green" title="E31">A9</font></i></a>, <a class="txt" href="weierstr.html#E14:31"><i><font color="Green" title="E32">A10</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/>
<a NAME="E17:31"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c4">p2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c3">p1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c3">p1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c3">p1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="weierstr.html#E13:31"><i><font color="Green" title="E31">A9</font></i></a>, <a class="txt" href="weierstr.html#E15:31"><i><font color="Green" title="E33">A11</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/>
<a NAME="E18:31"/><b>then </b>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c4">p2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c3">p1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/>
<a NAME="E19:31"/><b>then </b>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c3">p1</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c4">p2</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E20:31"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K4" title="WEIERSTR:func.4">upper_bound</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">P</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><a href="weierstr.html#K4" title="WEIERSTR:func.4">upper_bound</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">P</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>
 <b>by </b><i><a class="txt" href="weierstr.html#E10:31"><i><font color="Green" title="E28">A6</font></i></a>, <a class="txt" href="weierstr.html#E12:31"><i><font color="Green" title="E30">A8</font></i></a>, <a class="txt" href="weierstr.html#E16:31"><i><font color="Green" title="E34">A12</font></i></a>, <a class="ref" href="absvalue.html#T12" title="ABSVALUE:th.12">ABSVALUE:12</a></i>;<br/>


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