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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:9"/><i><font color="Green" title="E7">A1</font></i>: 
<font color="Maroon" title="c1">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E2:9"/><i><font color="Green" title="E8">A2</font></i>: 
<font color="Maroon" title="c2">B</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">A</font>
 ;<br/>

<a NAME="E3:9"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E2:9"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="sprect_1.html#T7" title="SPRECT_1:th.7">SPRECT_1:7</a></i>;<br/>
<a NAME="E4:9"/><i><font color="Green" title="E10">A4</font></i>: 
 <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#D8" title="EUCLID:def.8">EUCLID:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">C</font> = <font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c4">D</font> = <font color="Maroon" title="c1">A</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> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span></span>)</span> <b>by </b><i><a class="ref" href="euclid.html#D8" title="EUCLID:def.8">EUCLID:def 8</a></i>;<br/>
<a NAME="E6:9"/>
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</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> <font color="Maroon" title="c3">C</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:9_1"/><i><font color="Green" title="E11">A5</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">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="E3:9_1"/><i><font color="Green" title="E12">A6</font></i>: 
<font color="Maroon" title="c6">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>and </b><br/><a NAME="E4:9_1"/><i><font color="Green" title="E13">A7</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">D</font>
 <b>by </b><i><a class="txt" href="sprect_3.html#E1:9"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="sprect_3.html#E3:9"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="sprect_3.html#E4:9"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">r</font> = <font color="Maroon" title="c6">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/>
<b>take </b>
<font color="Maroon" title="c7">r</font>
;<br/>

<b>thus </b><a NAME="E6:9_1"/>
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="sprect_3.html#E3:9_1"><i><font color="Green" title="E12">A6</font></i></a></i>;<br/>

<b>reconsider </b><font color="Maroon" title="c8">E</font> =  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c7">r</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="sprect_3.html#E4:9"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></i>;<br/>
<a NAME="E8:9_1"/><i><font color="Green" title="E14">A8</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E3:9_1"><i><font color="Green" title="E12">A6</font></i></a>, <a class="ref" href="goboard6.html#T4" title="GOBOARD6:th.4">GOBOARD6:4</a></i>;<br/>
<a NAME="E9:9_1"/><b>then </b><i><font color="Green" title="E15">A9</font></i>: 
<font color="Maroon" title="c2">B</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c8">E</font>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:9_1"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E10:9_1"/>
<font color="Maroon" title="c8">E</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E8:9_1"><i><font color="Green" title="E14">A8</font></i></a>, <a class="ref" href="sprect_3.html#T17" target="_self" title="SPRECT_3:th.17">Th17</a></i>;<br/>
<b>hence </b><a NAME="E11:9_1"/>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c7">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font>
 <b>by </b><i><a class="txt" href="sprect_3.html#E2:9"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="sprect_3.html#E4:9_1"><i><font color="Green" title="E13">A7</font></i></a>, <a class="txt" href="sprect_3.html#E9:9_1"><i><font color="Green" title="E15">A9</font></i></a>, <a class="ref" href="goboard9.html#T6" title="GOBOARD9:th.6">GOBOARD9:6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:9"/>
<font color="Maroon" title="c2">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="sprect_3.html#E4:9"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a></i>;<br/>


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