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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<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>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</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>assume </b><a NAME="E1:18"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 ;<br/>

<a NAME="E2:18"/><i><font color="Green">E36</font></i>: 
 <a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:18_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">x</font> as   <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/>
<a NAME="E3:18_1"/><i><font color="Green">E39</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="metric_1.html#T13">METRIC_1:13</a></i>;<br/>
<a NAME="E4:18_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E5:18_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_4.html#E1"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E6:18_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:18"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> 1</span>)</span> is <a href="tbsp_1.html#V6">bounded</a>
 
<b>by </b><i><a class="ref" href="tbsp_1.html#T19">TBSP_1:19</a></i>;<br/>
<a NAME="E4:18"/><b>then </b>
 <a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> is <a href="tbsp_1.html#V6">bounded</a>
 
<b>by </b><i>, <a class="ref" href="tbsp_1.html#T21">TBSP_1:21</a></i>;<br/>
<b>hence </b><a NAME="E5:18"/>
<font color="Maroon">B</font> is <a href="jordan2c.html#V1">Bounded</a>
 <b>by </b><i>, <a class="ref" href="jordan2c.html#D2">JORDAN2C:def 2</a></i>;<br/>

<b>thus </b><a NAME="E6:18"/>
<font color="Maroon">B</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, <a class="ref" href="topreal6.html#T65">TOPREAL6:65</a></i>;<br/>


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