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<b>let </b><font color="Maroon">M</font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#NM1">MetrSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>;<br/>





<b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:94_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 ;<br/><b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E2:94_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>reconsider </b><font color="Maroon">B</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/><a NAME="E4:94_1"/>
<font color="Maroon">B</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p'</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:94_1"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="connsp_2.html#T34">CONNSP_2:34</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:94"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Olive">r</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E4:94"/>
for <font color="Olive">G</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p'</font>  holds <font color="Olive">G</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p'</font>;<br/>

<a NAME="E1:94_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:94_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:94_2"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:94_2"/>
<font color="Maroon">G</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:94"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="connsp_2.html#T34">CONNSP_2:34</a></i>;<br/>


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