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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">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>, <font color="Maroon">B</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>assume </b><a NAME="E1:77"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">A</font> is <a href="complsp1.html#V1" target="_self">open</a> &amp; <font color="Maroon">B</font> is <a href="complsp1.html#V1" target="_self">open</a> )
 ;<br/>

<b>let </b><font color="Maroon">C</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>assume </b><a NAME="E2:77"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font>
 ;<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="E3:77"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E4:77"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="complsp1.html#E45"><i><font color="Green">Lemma63</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:77"/><i><font color="Green">E103</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font>
 <b>and </b><br/><a NAME="E7:77"/><i><font color="Green">E109</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">r1</font> holds <br/><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><a class="ref" href="complsp1.html#T26" target="_self">Th26</a>, <a class="ref" href="complsp1.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E8:77"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="txt" href="complsp1.html#E45"><i><font color="Green">Lemma63</font></i></a>, <a class="ref" href="complsp1.html#T28" target="_self">Th28</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E10:77"/><i><font color="Green">E110</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font>
 <b>and </b><br/><a NAME="E11:77"/><i><font color="Green">E111</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">r2</font> holds <br/><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">B</font>
 <b>by </b><i><a class="ref" href="complsp1.html#T26" target="_self">Th26</a>, <a class="ref" href="complsp1.html#T24" target="_self">Th24</a></i>;<br/>
<b>take </b>
 <a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>
;<br/>

<b>thus </b><a NAME="E12:77"/>
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>
 <b>by </b><i><a class="ref" href="complsp1.html#T29" target="_self">Th29</a>, <a class="txt" href="complsp1.html#E49"><i><font color="Green">Lemma67</font></i></a>, <a class="ref" href="xxreal_0.html#T15">XXREAL_0:15</a></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/>

<a NAME="E13:77"/><i><font color="Green">E112</font></i>: 
(  <a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> &amp;  <a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/>
<b>assume </b><a NAME="E14:77"/>
<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>  <a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>
 ;<br/>

<a NAME="E15:77"/><b>then </b>
( <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">r1</font> &amp; <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">r2</font> )
 
<b>by </b><i><a class="ref" href="complsp1.html#T31" target="_self">Th31</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E16:77"/><b>then </b>
( <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> &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">B</font> )
 
<b>by </b><i><a class="txt" href="complsp1.html#E48"><i><font color="Green">Lemma66</font></i></a>, <a class="ref" href="complsp1.html#T30" target="_self">Th30</a></i>;<br/>
<b>hence </b><a NAME="E17:77"/>
<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">C</font>
 <b>by </b><i><a class="txt" href="complsp1.html#E45"><i><font color="Green">Lemma63</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


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