<?xml version="1.0"?>
<div class="add">

<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">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/>



<a NAME="E1:26"/>
for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>   st <font color="Olive" title="b1">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/><span class="p1">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ;<br/>

<b>assume </b><a NAME="E1:26_1"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c3">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<b>thus </b><a NAME="E2:26_1"/>
<span class="p1">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c3">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:26_1_1"/>
for <font color="Olive" title="b1">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 <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c3">P</font> holds <br/> ex <font color="Olive" title="b2">r</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 <br/>( <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c3">P</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">p</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><a NAME="E1:26_1_1_1"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c3">P</font>
 ;<br/>

<a NAME="E2:26_1_1_1"/><b>then </b><i><font color="Green" title="E18">A2</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c2">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a> <b> such that </b><br/><a NAME="E4:26_1_1_1"/><i><font color="Green" title="E19">A3</font></i>: 
<font color="Maroon" title="c5">y</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="c4">p</font>,<font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">P</font> = <font color="Maroon" title="c3">P</font> as   <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> <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a> </span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>;<br/>
<a NAME="E6:26_1_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 
<b>by </b><i><a class="txt" href="weierstr.html#E2:26_1_1_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="weierstr.html#E4:26_1_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="weierstr.html#D6" target="_self" title="WEIERSTR:def.6">Def6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">r</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> <b> such that </b><br/><a NAME="E8:26_1_1_1"/><i><font color="Green" title="E20">A4</font></i>: 
( <font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c7">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P</font> )
 <b>by </b><i><a class="txt" href="weierstr.html#E1:26_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">r</font> = <font color="Maroon" title="c7">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E10:26_1_1_1"/><i><font color="Green" title="E21">A5</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c8">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c6">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:26_1_1_1_1"/><i><font color="Green" title="E22">A6</font></i>: 
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c8">r</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c10">z</font> = <font color="Maroon" title="c9">z</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> ;<br/>
<a NAME="E3:26_1_1_1_1"/><i><font color="Green" title="E23">A7</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c10">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r</font>
 
<b>by </b><i><a class="txt" href="weierstr.html#E1:26_1_1_1_1"><i><font color="Green" title="E22">A6</font></i></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<a NAME="E4:26_1_1_1_1"/>
 <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="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">x</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="c10">z</font>,<font color="Maroon" title="c2">x</font></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="c4">p</font>,<font color="Maroon" title="c10">z</font>
 
<b>by </b><i><a class="ref" href="metric_6.html#T1" title="METRIC_6:th.1">METRIC_6:1</a></i>;<br/>
<a NAME="E5:26_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">z</font>,<font color="Maroon" title="c2">x</font></span>)</span></span>)</span></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">p</font>,<font color="Maroon" title="c10">z</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r</font> <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">p</font>,<font color="Maroon" title="c10">z</font></span>)</span>
 
<b>by </b><i><a class="txt" href="weierstr.html#E3:26_1_1_1_1"><i><font color="Green" title="E23">A7</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E6:26_1_1_1_1"/><b>then </b><i><font color="Green" title="E24">A8</font></i>: 
 <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="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">x</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="c10">z</font>,<font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a> <b> such that </b><br/><a NAME="E8:26_1_1_1_1"/><i><font color="Green" title="E25">A9</font></i>: 
<font color="Maroon" title="c11">q</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="c10">z</font>,<font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E9:26_1_1_1_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c11">q</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r</font>
 
<b>by </b><i><a class="txt" href="weierstr.html#E4:26_1_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="weierstr.html#E6:26_1_1_1_1"><i><font color="Green" title="E24">A8</font></i></a>, <a class="txt" href="weierstr.html#E8:26_1_1_1_1"><i><font color="Green" title="E25">A9</font></i></a>, <a class="ref" href="topmetr.html#T15" title="TOPMETR:th.15">TOPMETR:15</a></i>;<br/>
<a NAME="E10:26_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c8">r</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<a NAME="E11:26_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 
<b>by </b><i><a class="txt" href="weierstr.html#E8:26_1_1_1"><i><font color="Green" title="E20">A4</font></i></a></i>;<br/>
<a NAME="E12:26_1_1_1_1"/><b>then </b><i><font color="Green" title="E26">A10</font></i>: 
<span class="p1">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 
<b>by </b><i><a class="txt" href="weierstr.html#E8:26_1_1_1_1"><i><font color="Green" title="E25">A9</font></i></a>, <a class="ref" href="weierstr.html#D6" target="_self" title="WEIERSTR:def.6">Def6</a></i>;<br/>
<a NAME="E13:26_1_1_1_1"/>
 <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="c2">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="E14:26_1_1_1_1"/><b>then </b>
 <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="c2">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 <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/>
<b>hence </b><a NAME="E15:26_1_1_1_1"/>
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c6">P</font>
 <b>by </b><i><a class="txt" href="weierstr.html#E12:26_1_1_1_1"><i><font color="Green" title="E26">A10</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon" title="c8">r</font>
;<br/>

<b>thus </b><a NAME="E11:26_1_1_1"/>
( <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c8">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c3">P</font> )
 <b>by </b><i><a class="txt" href="weierstr.html#E8:26_1_1_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="weierstr.html#E10:26_1_1_1"><i><font color="Green" title="E21">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:26_1_1"/>
<span class="p1">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c3">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:26"/>
 <a href="weierstr.html#K6" title="WEIERSTR:func.6">dist</a> <font color="Maroon" title="c2">x</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>


</div>
