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<b>let </b><font color="Maroon">X</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<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">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" target="_self">TopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:35"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Olive">y</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> </span>}</span> 
 ;<br/>

<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K2" target="_self">MetricSpaceNorm</a> <font color="Maroon">X</font></span>)</span> ;<br/>
<a NAME="E3:35"/>
 ex <font color="Olive">t</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">z</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">t</font> <a href="rlvect_1.html#K6">-</a> <font color="Olive">y</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> </span>}</span>  )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:35"/>
<font color="Maroon">V</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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