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<div class="add">

<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">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">n</font></span>)</span>;<br/>



<b>thus </b><a NAME="E1:112"/>
( <font color="Maroon">A</font> is <a href="complsp1.html#V2" target="_self">closed</a> implies <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="complsp1.html#V1" target="_self">open</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:112_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">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>  st ( for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">z</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> st <br/>( <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><font color="Olive">z</font></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">x</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) ) holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="complsp1.html#D16" target="_self">COMPLSP1:def 16</a><br/>

<b>let </b><font color="Maroon">x</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">n</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="complsp1.html#D15" target="_self">COMPLSP1:def 15</a><br/>

<b>assume </b><a NAME="E2:112_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<a NAME="E3:112_1"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:112_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E6:112_1"/><i><font color="Green">E102</font></i>: 
for <font color="Olive">z</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>  st <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><font color="Olive">z</font></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> holds <br/> not <font color="Maroon">x</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">r</font>
;<br/>

<b>thus </b><a NAME="E7:112_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">z</font> be    <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>assume </b><a NAME="E8:112_1"/>
<span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/>

<a NAME="E9:112_1"/><b>then </b>
not <font color="Maroon">x</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="complsp1.html#T59" target="_self">Th59</a></i>;<br/>
<b>hence </b><a NAME="E10:112_1"/>
<font color="Maroon">x</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:112"/><i><font color="Green">E103</font></i>: 
for <font color="Olive">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>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  holds <br/> ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> &amp; ( for <font color="Olive">z</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>  st <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><font color="Olive">z</font></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> holds <br/><font color="Olive">x</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="complsp1.html#D15" target="_self">COMPLSP1:def 15</a><br/>

<b>let </b><font color="Maroon">x</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">n</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="complsp1.html#D16" target="_self">COMPLSP1:def 16</a><br/>

<b>assume </b><a NAME="E3:112"/><i><font color="Green">E109</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">z</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> st <br/>( <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><font color="Olive">z</font></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> &amp; <font color="Maroon">x</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:112_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>then consider </b><font color="Maroon">z</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="E3:112_2"/><i><font color="Green">E110</font></i>: 
( <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">x</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">z</font> = <font color="Maroon">z</font>;<br/><b>thus </b><a NAME="E4:112_2"/>
( <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp; not <font color="Maroon">x</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  )
 <b>by </b><i><a class="ref" href="complsp1.html#T59" target="_self">Th59</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:112"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:112"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/>


</div>
