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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">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">x</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>assume </b><a NAME="E1:20"/>
not  <a href="topreal9.html#K1" target="_self">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal9.html#K3" target="_self">Sphere</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 ;<br/>

<b>then consider </b><font color="Maroon">q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:20"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1" target="_self">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E4:20"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K3" target="_self">Sphere</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as   <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> <b>by </b><i/>;<br/>
<a NAME="E6:20"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:20"/>
contradiction
 <b>by </b><i>, </i>;<br/>


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