<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <a href="csspace.html#K20">Complex_l2_Space</a> ;<br/>

<a NAME="E1:17"/><i><font color="Green">E18</font></i>: 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="ref" href="csspace.html#D11">CSSPACE:def 11</a>, <a class="ref" href="csspace.html#D20">CSSPACE:def 20</a></i>;<br/>
<a NAME="E2:17"/><i><font color="Green">E19</font></i>: 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="ref" href="csspace.html#D11">CSSPACE:def 11</a>, <a class="ref" href="csspace.html#D20">CSSPACE:def 20</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="seq_1.html#K22">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span>;<br/>
<a NAME="E3:17"/><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="txt" href="csspace2.html#E1:17"><i><font color="Green">E18</font></i></a>, <a class="txt" href="csspace2.html#E2:17"><i><font color="Green">E19</font></i></a>, <a class="ref" href="series_1.html#T10">SERIES_1:10</a></i>;<br/>
<a NAME="E4:17"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span>;<br/>
<i><font color="Green">E22</font></i>: <span class="p1">(<span class="default">2 <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> = 
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
<br/>.= 
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> =  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E3:17_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> 2 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<b>hence </b><a NAME="E4:17_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="csspace2.html#E1:17_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xreal_1.html#T129">XREAL_1:129</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:17"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default">2 <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span>, <font color="Maroon">c<sub>10</sub></font> = <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> =  <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>12</sub></font> =  <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>10</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<i><font color="Green">E22</font></i>: <span class="p1">(<span class="default">2 <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> = 
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13">SEQ_1:13</a></i>
<br/>.= 
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>
<br/>.= 
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>
<br/>.= 
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
<br/>.= 
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>
<br/>.= 
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>

;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>

;<br/>
<b>then </b><i><font color="Green">E23</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p4">(<span class="default"><span class="p5"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p5"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="csspace2.html#E3:17_2"><i><font color="Green">E22</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 

;<br/>
<a NAME="E6:17_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E7:17_2"/><b>then </b>
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p4">(<span class="default"><span class="p5"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p5"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p5"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p4">(<span class="default"><span class="p5"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p5"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="seq_1.html#K14">(#)</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p5">(<span class="default"><span class="p0"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p0"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p1">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="seq_1.html#K14">(#)</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p4">(<span class="default"><span class="p5"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p5"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="csspace2.html#E5:17_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E8:17_2"/>
<span class="p1">(<span class="default">2 <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:17"/><b>then </b><i><font color="Green">E22</font></i>: 
2 <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="txt" href="csspace2.html#E3:17"><i><font color="Green">E20</font></i></a>, <a class="txt" href="csspace2.html#E4:17"><i><font color="Green">E21</font></i></a>, <a class="ref" href="series_1.html#T24">SERIES_1:24</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = 1 <a href="real_1.html#K6">/</a> 2;<br/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default">2 <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3">(<span class="default"><span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p4"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> 2</span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2">(<span class="default"><span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T31">SEQ_1:31</a></i>
<br/>.= 
 <a href="seq_1.html#K22">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T35">SEQ_1:35</a></i>
;<br/>
<a NAME="E8:17"/><b>then </b>
 <a href="seq_1.html#K22">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="txt" href="csspace2.html#E6:17"><i><font color="Green">E22</font></i></a>, <a class="ref" href="series_1.html#T13">SERIES_1:13</a></i>;<br/>
<a NAME="E9:17"/><b>then </b>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V2">absolutely_summable</a>
 
<b>by </b><i><a class="ref" href="series_1.html#D5">SERIES_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E10:17"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K11">(#)</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> is <a href="series_1.html#V1">summable</a>
 <b>by </b><i><a class="ref" href="series_1.html#T40">SERIES_1:40</a></i>;<br/>


</div>
