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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">S</font> be   <a href="prob_1.html#NM2">SetSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">p'</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:68"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:68_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">S</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="E2:68_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>then reconsider </b><font color="Maroon">G</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p</font> <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:68_1"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">S</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="kurato_2.html#E1:68"><i><font color="Green">E43</font></i></a>, </i>;<br/><b>thus </b><a NAME="E6:68_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">k</font> holds <br/><font color="Maroon">S</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="kurato_2.html#T3" target="_self">Th3</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:68"/><i><font color="Green">E56</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/> ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">k</font> holds <br/><font color="Maroon">S</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Olive">r</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2"><b>now </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/><b>reconsider </b><font color="Maroon">G'</font> =  <a href="tops_1.html#K1">Int</a> <font color="Maroon">G</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> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span></span>)</span> ;<br/><a NAME="E2:68_2"/>
<font color="Maroon">p'</font> <a href="hidden.html#R2">in</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="E4:68_2"/><i><font color="Green">E57</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><a class="ref" href="topmetr.html#T22">TOPMETR:22</a></i>;<br/><a NAME="E5:68_2"/>
<font color="Maroon">G'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/><a NAME="E6:68_2"/><b>then </b><i><font color="Green">E72</font></i>: 
 <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><a class="ref" href="kurato_2.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:68_2"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">S</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="kurato_2.html#E1:68"><i><font color="Green">E43</font></i></a>, <a class="ref" href="kurato_2.html#T3" target="_self">Th3</a></i>;<br/><b>take </b><font color="Maroon">k</font> = <font color="Maroon">k</font>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E9:68_2"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E10:68_2"/><b>then </b>
<font color="Maroon">S</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E11:68_2"/>
<font color="Maroon">S</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</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="E5:68"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">S</font>
 <b>by </b><i/>;<br/>


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