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

<b>let </b><font color="Maroon" title="c1">u</font>, <font color="Maroon" title="c2">v</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <a href="csspace.html#K18" title="CSSPACE:func.18">Complex_l2_Space</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a></span><br/>

<a NAME="E1:17"/><i><font color="Green" title="E18">A1</font></i>: 
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 
<b>by </b><i><a class="ref" href="csspace.html#D11" title="CSSPACE:def.11">CSSPACE:def 11</a>, <a class="ref" href="csspace.html#D20" title="CSSPACE:def.20">CSSPACE:def 20</a></i>;<br/>
<a NAME="E2:17"/><i><font color="Green" title="E19">A2</font></i>: 
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 
<b>by </b><i><a class="ref" href="csspace.html#D11" title="CSSPACE:def.11">CSSPACE:def 11</a>, <a class="ref" href="csspace.html#D20" title="CSSPACE:def.20">CSSPACE:def 20</a></i>;<br/>
<b>set </b><font color="Maroon" title="c3">p</font> = <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span>;<br/>
<b>set </b><font color="Maroon" title="c4">q</font> = <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span>;<br/>
<b>set </b><font color="Maroon" title="c5">r</font> =  <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span>;<br/>
<a NAME="E3:17"/><i><font color="Green" title="E20">A3</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 
<b>by </b><i><a class="txt" href="#E1:17"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="#E2:17"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="series_1.html#T10" title="SERIES_1:th.10">SERIES_1:10</a></i>;<br/>
<a NAME="E4:17"/><i><font color="Green" title="E21">A4</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p3">(<span class="default"><span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</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" title="c6">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p3">(<span class="default"><span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span><br/>

<b>set </b><font color="Maroon" title="c7">rr</font> = <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span>;<br/>
<i><font color="Green" title="E22">A5</font></i>: <span class="p1">(<span class="default">2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p3">(<span class="default"><span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> = 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p3">(<span class="default"><span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
<br/>.= 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T16" title="SEQ_1:th.16">SEQ_1:16</a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c8">tt</font> =  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E3:17_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">tt</font> )
 
<b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/>
<b>hence </b><a NAME="E4:17_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p3">(<span class="default"><span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="#E1:17_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="xreal_1.html#T129" title="XREAL_1:th.129">XREAL_1:129</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:17"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default">2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p3">(<span class="default"><span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <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"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</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" title="c6">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default">2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p3">(<span class="default"><span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <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"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span><br/>

<b>set </b><font color="Maroon" title="c7">s</font> =  <a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font>;<br/>
<b>set </b><font color="Maroon" title="c8">t</font> =  <a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">sn</font> = <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>, <font color="Maroon" title="c10">tn</font> = <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c11">ss</font> =  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">sn</font>, <font color="Maroon" title="c12">tt</font> =  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c10">tn</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<i><font color="Green" title="E22">A6</font></i>: <span class="p1">(<span class="default">2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p3">(<span class="default"><span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> = 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p3">(<span class="default"><span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
<br/>.= 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T16" title="SEQ_1:th.16">SEQ_1:16</a></i>
<br/>.= 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>
<br/>.= 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T151" title="COMPLEX1:th.151">COMPLEX1:151</a></i>
<br/>.= 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p3">(<span class="default"><span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#T18" title="VALUED_1:th.18">VALUED_1:18</a></i>
<br/>.= 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">ss</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c12">tt</font></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#T18" title="VALUED_1:th.18">VALUED_1:18</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">sn</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c10">tn</font></span>)</span>

;<br/>
<i><font color="Green" title="E23">A7</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#T18" title="VALUED_1:th.18">VALUED_1:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">sn</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#T18" title="VALUED_1:th.18">VALUED_1:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">sn</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#T18" title="VALUED_1:th.18">VALUED_1:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">sn</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c10">tn</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#T18" title="VALUED_1:th.18">VALUED_1:18</a></i>
;<br/>
<a NAME="E5:17_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c9">sn</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c10">tn</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></i>;<br/>
<a NAME="E6:17_2"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p4">(<span class="default"><span class="p5"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p4">(<span class="default"><span class="p5"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p5">(<span class="default"><span class="p0"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p0"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p4">(<span class="default"><span class="p5"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p0">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:17_2"><i><font color="Green" title="E22">A6</font></i></a>, <a class="txt" href="#E4:17_2"><i><font color="Green" title="E23">A7</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E7:17_2"/>
<span class="p1">(<span class="default">2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p3">(<span class="default"><span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <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"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E6:17"/><b>then </b><i><font color="Green" title="E22">A8</font></i>: 
2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span></span>)</span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 
<b>by </b><i><a class="txt" href="#E3:17"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="#E4:17"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="series_1.html#T24" title="SERIES_1:th.24">SERIES_1:24</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">q0</font> = 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2;<br/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default">2 <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p3">(<span class="default"><span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p5">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</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#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 2</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p2">(<span class="default"><span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p4">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T31" title="SEQ_1:th.31">SEQ_1:31</a></i>
<br/>.= 
 <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T35" title="SEQ_1:th.35">SEQ_1:35</a></i>
;<br/>
<a NAME="E8:17"/><b>then </b>
 <a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 
<b>by </b><i><a class="txt" href="#E6:17"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="series_1.html#T13" title="SERIES_1:th.13">SERIES_1:13</a></i>;<br/>
<a NAME="E9:17"/><b>then </b>
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is <a href="series_1.html#V2" title="SERIES_1:attr.2">absolutely_summable</a>
 
<b>by </b><i><a class="ref" href="series_1.html#D5" title="SERIES_1:def.5">SERIES_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E10:17"/>
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c1">u</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c2">v</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
