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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="complsp1.html#K23" title="COMPLSP1:func.23">the_Complex_Space</a> <font color="Maroon" title="c1">n</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D5" title="COMPTS_1:def.5">COMPTS_1:def 5</a><br/><b>let </b><font color="Maroon" title="c3">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="complsp1.html#K23" title="COMPLSP1:func.23">the_Complex_Space</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:115"/><i><font color="Green" title="E72">A1</font></i>: 
( <font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">P</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">z1</font> = <font color="Maroon" title="c2">p</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c3">P</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<a NAME="E4:115"/><i><font color="Green" title="E73">A2</font></i>: 
<font color="Maroon" title="c5">A</font> is <a href="complsp1.html#V2" title="COMPLSP1:attr.2">closed</a>
 
<b>by </b><i><a class="txt" href="complsp1.html#E1:115"><i><font color="Green" title="E72">A1</font></i></a>, <a class="ref" href="complsp1.html#T110" target="_self" title="COMPLSP1:th.110">Th110</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">d</font> = <span class="p1">(<span class="default"><a href="complsp1.html#K18" title="COMPLSP1:func.18">dist</a> <font color="Maroon" title="c4">z1</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2;<br/>
<a NAME="E5:115"/>
not <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="txt" href="complsp1.html#E1:115"><i><font color="Green" title="E72">A1</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E6:115"/><b>then </b>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="complsp1.html#K18" title="COMPLSP1:func.18">dist</a> <font color="Maroon" title="c4">z1</font>,<font color="Maroon" title="c5">A</font>
 
<b>by </b><i><a class="txt" href="complsp1.html#E1:115"><i><font color="Green" title="E72">A1</font></i></a>, <a class="txt" href="complsp1.html#E4:115"><i><font color="Green" title="E73">A2</font></i></a>, <a class="ref" href="complsp1.html#T88" target="_self" title="COMPLSP1:th.88">Th88</a></i>;<br/>
<a NAME="E7:115"/><b>then </b><i><font color="Green" title="E74">A3</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="complsp1.html#K18" title="COMPLSP1:func.18">dist</a> <font color="Maroon" title="c4">z1</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T217" title="XREAL_1:th.217">XREAL_1:217</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">K1</font> =  <a href="complsp1.html#K17" title="COMPLSP1:func.17">Ball</a> <font color="Maroon" title="c4">z1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complsp1.html#K18" title="COMPLSP1:func.18">dist</a> <font color="Maroon" title="c4">z1</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span>, <font color="Maroon" title="c8">K2</font> =  <a href="complsp1.html#K19" title="COMPLSP1:func.19">Ball</a> <font color="Maroon" title="c5">A</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complsp1.html#K18" title="COMPLSP1:func.18">dist</a> <font color="Maroon" title="c4">z1</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="complsp1.html#K23" title="COMPLSP1:func.23">the_Complex_Space</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c7">K1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c8">K2</font>
;<br/>

<a NAME="E9:115"/>
(  <a href="complsp1.html#K17" title="COMPLSP1:func.17">Ball</a> <font color="Maroon" title="c4">z1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complsp1.html#K18" title="COMPLSP1:func.18">dist</a> <font color="Maroon" title="c4">z1</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> is <a href="complsp1.html#V1" title="COMPLSP1:attr.1">open</a> &amp;  <a href="complsp1.html#K19" title="COMPLSP1:func.19">Ball</a> <font color="Maroon" title="c5">A</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complsp1.html#K18" title="COMPLSP1:func.18">dist</a> <font color="Maroon" title="c4">z1</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> is <a href="complsp1.html#V1" title="COMPLSP1:attr.1">open</a> )
 
<b>by </b><i><a class="txt" href="complsp1.html#E1:115"><i><font color="Green" title="E72">A1</font></i></a>, <a class="ref" href="complsp1.html#T82" target="_self" title="COMPLSP1:th.82">Th82</a>, <a class="ref" href="complsp1.html#T93" target="_self" title="COMPLSP1:th.93">Th93</a></i>;<br/>
<b>hence </b><a NAME="E10:115"/>
( <font color="Maroon" title="c7">K1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c8">K2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="ref" href="complsp1.html#T108" target="_self" title="COMPLSP1:th.108">Th108</a></i>;<br/>

<b>thus </b><a NAME="E11:115"/>
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">K1</font> &amp; <font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">K2</font> )
 <b>by </b><i><a class="txt" href="complsp1.html#E7:115"><i><font color="Green" title="E74">A3</font></i></a>, <a class="ref" href="complsp1.html#T81" target="_self" title="COMPLSP1:th.81">Th81</a>, <a class="ref" href="complsp1.html#T92" target="_self" title="COMPLSP1:th.92">Th92</a></i>;<br/>

<b>assume </b><a NAME="E12:115"/>
<font color="Maroon" title="c7">K1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">K2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a><br/>

<b>then consider </b><font color="Maroon" title="c9">x</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E14:115"/><i><font color="Green" title="E75">A4</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="complsp1.html#K17" title="COMPLSP1:func.17">Ball</a> <font color="Maroon" title="c4">z1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complsp1.html#K18" title="COMPLSP1:func.18">dist</a> <font color="Maroon" title="c4">z1</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="complsp1.html#K19" title="COMPLSP1:func.19">Ball</a> <font color="Maroon" title="c5">A</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complsp1.html#K18" title="COMPLSP1:func.18">dist</a> <font color="Maroon" title="c4">z1</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/>
<a NAME="E15:115"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">K1</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">K2</font> )
 
<b>by </b><i><a class="txt" href="complsp1.html#E14:115"><i><font color="Green" title="E75">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E16:115"/><b>then </b>
( <span class="p1"><a href="complsp1.html#K16" title="COMPLSP1:func.16">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">z1</font> <a href="complsp1.html#K13" title="COMPLSP1:func.13">-</a> <font color="Maroon" title="c9">x</font></span>)</span></span><a href="complsp1.html#K16" title="COMPLSP1:func.16">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="complsp1.html#K18" title="COMPLSP1:func.18">dist</a> <font color="Maroon" title="c4">z1</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2 &amp;  <a href="complsp1.html#K18" title="COMPLSP1:func.18">dist</a> <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c5">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="complsp1.html#K18" title="COMPLSP1:func.18">dist</a> <font color="Maroon" title="c4">z1</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2 )
 
<b>by </b><i><a class="ref" href="complsp1.html#T80" target="_self" title="COMPLSP1:th.80">Th80</a>, <a class="ref" href="complsp1.html#T90" target="_self" title="COMPLSP1:th.90">Th90</a></i>;<br/>
<a NAME="E17:115"/><b>then </b>
<span class="p1"><a href="complsp1.html#K16" title="COMPLSP1:func.16">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">z1</font> <a href="complsp1.html#K13" title="COMPLSP1:func.13">-</a> <font color="Maroon" title="c9">x</font></span>)</span></span><a href="complsp1.html#K16" title="COMPLSP1:func.16">.|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="complsp1.html#K18" title="COMPLSP1:func.18">dist</a> <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complsp1.html#K18" title="COMPLSP1:func.18">dist</a> <font color="Maroon" title="c4">z1</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complsp1.html#K18" title="COMPLSP1:func.18">dist</a> <font color="Maroon" title="c4">z1</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E18:115"/>
contradiction
 <b>by </b><i><a class="txt" href="complsp1.html#E1:115"><i><font color="Green" title="E72">A1</font></i></a>, <a class="ref" href="complsp1.html#T89" target="_self" title="COMPLSP1:th.89">Th89</a></i>;<br/>


</div>
