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<span class="kw">reconsider </span><font color="Maroon" title="c4">e</font> = <font color="Maroon" title="c2">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal3.html#T8" title="TOPREAL3:th.8">TOPREAL3:8</a></span>;<br/>
<a NAME="E2:25_3_1"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E3:25_3_1"/><span class="kw">then </span>
(  <a href="metric_1.html#K11" title="METRIC_1:func.11">Sphere</a> (<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">r</font>) is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>  &amp;  <a href="metric_1.html#K11" title="METRIC_1:func.11">Sphere</a> (<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">r</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal9.html#T15" target="_self" title="TOPREAL9:th.15">Th15</a>, <a class="ref" href="topreal6.html#T62" title="TOPREAL6:th.62">TOPREAL6:62</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:25_3_1"/>
 <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font>) is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25_3_1"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="topreal6.html#T61" title="TOPREAL6:th.61">TOPREAL6:61</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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