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<b>assume </b><a NAME="E1:21_1_1"/>
not  <a href="topreal9.html#K1" target="_self">Ball</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:21_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1" target="_self">Ball</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<a NAME="E4:21_1_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K20">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="topreal9.html#E3:21_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="topreal9.html#T7" target="_self">Th7</a></i>;<br/>
<b>hence </b><a NAME="E5:21_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="toprns_1.html#T26">TOPRNS_1:26</a></i>;<br/>


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