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<b>let </b><font color="Maroon" title="c1">PM</font> be    <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">PM</font>;<br/>



<b>assume </b><a NAME="E1:37"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <font color="Maroon" title="c1">PM</font>
 ;<br/>

<a NAME="E2:37"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">PM</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">A</font> holds <br/> ex <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">A</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">PM</font>;<br/>

<b>assume </b><a NAME="E1:37_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">W</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:37_1"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">W</font>
 <b>and </b><br/><a NAME="E4:37_1"/><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon" title="c4">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">W</font> = <font color="Maroon" title="c4">W</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">PM</font> <b>by </b><i><a class="txt" href="pcomps_1.html#E4:37_1"><i><font color="Green" title="E31">A3</font></i></a></i>;<br/>
<a NAME="E6:37_1"/><i><font color="Green" title="E32">A4</font></i>: 
<font color="Maroon" title="c5">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="pcomps_1.html#E4:37_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E8:37_1"/><i><font color="Green" title="E33">A5</font></i>: 
( <font color="Maroon" title="c6">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">W</font> )
 <b>by </b><i><a class="txt" href="pcomps_1.html#E1:37"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="pcomps_1.html#E3:37_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="pcomps_1.html#E4:37_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="pcomps_1.html#D5" target="_self" title="PCOMPS_1:def.5">Def5</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">r</font>
;<br/>

<b>thus </b><a NAME="E9:37_1"/>
<font color="Maroon" title="c6">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="pcomps_1.html#E8:37_1"><i><font color="Green" title="E33">A5</font></i></a></i>;<br/>

<b>thus </b><a NAME="E10:37_1"/>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E6:37_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="pcomps_1.html#E8:37_1"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:37"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <font color="Maroon" title="c1">PM</font>
 <b>by </b><i><a class="ref" href="pcomps_1.html#D5" target="_self" title="PCOMPS_1:def.5">Def5</a></i>;<br/>


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