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

<span class="kw">let </span><font color="Maroon" title="c3">z</font> be   <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">z</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K28" title="SIN_COS:func.28">P_dt</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">z</font></span><br/>

<a NAME="E1:94"/><span class="lab"><font color="Green" title="E76">A1</font></span>: 
 for <font color="Olive" title="b1">z</font> being   <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Olive" title="b1">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">z</font> <a href="sin_cos.html#K28" title="SIN_COS:func.28">P_dt</a></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">z</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">z</font> be   <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="sin_cos.html#K28" title="SIN_COS:func.28">P_dt</a></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 2</span><br/>

<a NAME="E1:94_1"/>
 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"><font color="Maroon" title="c4">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="sin_cos.html#K28" title="SIN_COS:func.28">P_dt</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 2</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="sin_cos.html#K28" title="SIN_COS:func.28">P_dt</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 2</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font></span><br/>

<span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="sin_cos.html#K28" title="SIN_COS:func.28">P_dt</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font> = 
<font color="Maroon" title="c4">z</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="sin_cos.html#K28" title="SIN_COS:func.28">P_dt</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T6" title="VALUED_1:th.6">VALUED_1:6</a></span>
<br/>.= 
<font color="Maroon" title="c4">z</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#D24" target="_self" title="SIN_COS:def.24">Def28</a></span>
;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="sin_cos.html#K28" title="SIN_COS:func.28">P_dt</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="newton.html#K5" title="NEWTON:func.5">!</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T6" title="NEWTON:th.6">NEWTON:6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#D4" target="_self" title="SIN_COS:def.4">Def8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 2</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:94_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="sin_cos.html#K28" title="SIN_COS:func.28">P_dt</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 2</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:94_1"/>
<font color="Maroon" title="c4">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="sin_cos.html#K28" title="SIN_COS:func.28">P_dt</a></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" title="FUNCT_2:th.63">FUNCT_2:63</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
 <a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1</span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#T60" title="COMSEQ_3:th.60">COMSEQ_3:60</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1</span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> 2</span>)</span></span>)</span>

;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E77">A2</font></span>:  <a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K28" title="SIN_COS:func.28">P_dt</a></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:94"><span class="lab"><font color="Green" title="E76">A1</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="sin_cos.html#E77"><span class="lab"><font color="Green" title="E60">Lm8</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="sin_cos.html#E77"><span class="lab"><font color="Green" title="E60">Lm8</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">z</font>

;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">BBB</font> = <font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K28" title="SIN_COS:func.28">P_dt</a>  as   <a href="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a>  <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T56" target="_self" title="SIN_COS:th.56">Th61</a></span>;<br/>
<a NAME="E5:94"/>
<font color="Maroon" title="c4">BBB</font> is  <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E6:94"/>
<font color="Maroon" title="c3">z</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K28" title="SIN_COS:func.28">P_dt</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:94"><span class="lab"><font color="Green" title="E77">A2</font></span></a>, <a class="ref" href="comseq_3.html#T56" title="COMSEQ_3:th.56">COMSEQ_3:56</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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