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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="complsp1.html#K23" target="_self">the_Complex_Space</a> <font color="Maroon">n</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D4">COMPTS_1:def 4</a><br/><b>let </b><font color="Maroon">q</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="complsp1.html#K23" target="_self">the_Complex_Space</a> <font color="Maroon">n</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:114"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">z1</font> = <font color="Maroon">p</font>, <font color="Maroon">z2</font> = <font color="Maroon">q</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">n</font> ;<br/>
<b>set </b><font color="Maroon">d</font> = <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">z2</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="binop_2.html#K12">/</a> 2;<br/>
<a NAME="E3:114"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">z2</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span>
 
<b>by </b><i>, <a class="txt" href="complsp1.html#E34"><i><font color="Green">Lemma55</font></i></a></i>;<br/>
<a NAME="E4:114"/><b>then </b><i><font color="Green">E60</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">z2</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="binop_2.html#K12">/</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T217">XREAL_1:217</a></i>;<br/>
<b>reconsider </b><font color="Maroon">K1</font> =  <a href="complsp1.html#K17" target="_self">Ball</a> <font color="Maroon">z1</font>,<span class="p1">(<span class="default"><span class="p2"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">z2</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="binop_2.html#K12">/</a> 2</span>)</span>, <font color="Maroon">K2</font> =  <a href="complsp1.html#K17" target="_self">Ball</a> <font color="Maroon">z2</font>,<span class="p1">(<span class="default"><span class="p2"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">z2</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="binop_2.html#K12">/</a> 2</span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="complsp1.html#K23" target="_self">the_Complex_Space</a> <font color="Maroon">n</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">K1</font>
;<br/>

<b>take </b>
<font color="Maroon">K2</font>
;<br/>

<a NAME="E6:114"/>
(  <a href="complsp1.html#K17" target="_self">Ball</a> <font color="Maroon">z1</font>,<span class="p1">(<span class="default"><span class="p2"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">z2</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="binop_2.html#K12">/</a> 2</span>)</span> is <a href="complsp1.html#V1" target="_self">open</a> &amp;  <a href="complsp1.html#K17" target="_self">Ball</a> <font color="Maroon">z2</font>,<span class="p1">(<span class="default"><span class="p2"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">z2</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="binop_2.html#K12">/</a> 2</span>)</span> is <a href="complsp1.html#V1" target="_self">open</a> )
 
<b>by </b><i><a class="ref" href="complsp1.html#T28" target="_self">Th28</a></i>;<br/>
<b>hence </b><a NAME="E7:114"/>
( <font color="Maroon">K1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">K2</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="complsp1.html#T52" target="_self">Th52</a></i>;<br/>

<b>thus </b><a NAME="E8:114"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K1</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K2</font> )
 <b>by </b><i><a class="ref" href="complsp1.html#T59" target="_self">Th59</a>, <a class="txt" href="complsp1.html#E45"><i><font color="Green">Lemma63</font></i></a></i>;<br/>

<b>assume </b><a NAME="E9:114"/>
<font color="Maroon">K1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E11:114"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="complsp1.html#K17" target="_self">Ball</a> <font color="Maroon">z1</font>,<span class="p2">(<span class="default"><span class="p3"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">z2</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="binop_2.html#K12">/</a> 2</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="complsp1.html#K17" target="_self">Ball</a> <font color="Maroon">z2</font>,<span class="p2">(<span class="default"><span class="p3"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">z2</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="binop_2.html#K12">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<a NAME="E12:114"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K2</font> )
 
<b>by </b><i><a class="ref" href="complsp1.html#T60" target="_self">Th60</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E13:114"/><b>then </b>
( <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">x</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">z2</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="binop_2.html#K12">/</a> 2 &amp; <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">x</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">z2</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="binop_2.html#K12">/</a> 2 )
 
<b>by </b><i><a class="ref" href="complsp1.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E14:114"/><b>then </b>
<span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">x</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="binop_2.html#K9">+</a> <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">x</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">z2</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">z2</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="binop_2.html#K12">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E15:114"/><b>then </b>
<span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">x</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="binop_2.html#K9">+</a> <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">z2</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">z2</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span>
 
<b>by </b><i><a class="ref" href="complsp1.html#T21" target="_self">Th21</a></i>;<br/>
<b>hence </b><a NAME="E16:114"/>
contradiction
 <b>by </b><i><a class="ref" href="complsp1.html#T23" target="_self">Th23</a></i>;<br/>


</div>
