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

<b>let </b><font color="Maroon" title="c3">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/>

<a NAME="E1:103"/><i><font color="Green" title="E86">A1</font></i>: 
for <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  holds  <font color="Olive" title="b1">z</font> <a href="comseq_1.html#K4" title="COMSEQ_1:func.4">(#)</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">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#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/>

<a NAME="E1:103_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="comseq_1.html#K4" title="COMSEQ_1:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</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#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> 2</span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_1"><b>proof </b></a><div class="add">

<b>let </b><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> ;<br/>

<span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="comseq_1.html#K4" title="COMSEQ_1:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font> = 
<font color="Maroon" title="c4">z</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</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#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D7" title="COMSEQ_1:def.7">COMSEQ_1:def 7</a></i>
<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="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</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="sin_cos.html#K4" title="SIN_COS:func.4">!c</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D28" target="_self" title="SIN_COS:def.28">Def28</a></i>
;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="comseq_1.html#K4" title="COMSEQ_1:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</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="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</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="sin_cos.html#K4" title="SIN_COS:func.4">!c</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></i>
<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="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</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="sin_cos.html#K4" title="SIN_COS:func.4">!c</a> </span>)</span>
<b>by </b><i><a class="txt" href="sin_cos.html#E90"><i><font color="Green" title="E70">Lm13</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</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>
<b>by </b><i><a class="ref" href="sin_cos.html#D8" target="_self" title="SIN_COS:def.8">Def8</a></i>
<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#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> 2</span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font>
<b>by </b><i><a class="ref" href="comseq_3.html#D6" title="COMSEQ_3:def.6">COMSEQ_3:def 6</a></i>
;<br/>
<b>hence </b><a NAME="E3:103_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="comseq_1.html#K4" title="COMSEQ_1:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</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#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> 2</span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:103_1"/>
<font color="Maroon" title="c4">z</font> <a href="comseq_1.html#K4" title="COMSEQ_1:func.4">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">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#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> 2
 <b>by </b><i><a class="ref" href="comseq_1.html#T6" title="COMSEQ_1:th.6">COMSEQ_1:6</a></i>;<br/>


</div><b>end;</b></div>
 <a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</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#K8" title="COMSEQ_3:func.8">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#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</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>
<b>by </b><i><a class="ref" href="comseq_3.html#T61" title="COMSEQ_3:th.61">COMSEQ_3:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</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#K8" title="COMSEQ_3:func.8">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#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span> <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">^\</a> 2</span>)</span></span>)</span>

;<br/>
<b>then </b><i><font color="Green" title="E87">A2</font></i>:  <a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="comseq_1.html#K4" title="COMSEQ_1:func.4">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">P_dt</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">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="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</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>
<b>by </b><i><a class="txt" href="sin_cos.html#E1:103"><i><font color="Green" title="E86">A1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">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="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</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#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D7" title="COMSEQ_3:def.7">COMSEQ_3:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">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#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</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#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D7" title="COMSEQ_3:def.7">COMSEQ_3:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">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#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sin_cos.html#E90"><i><font color="Green" title="E70">Lm13</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">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> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="txt" href="sin_cos.html#E90"><i><font color="Green" title="E70">Lm13</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">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="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">z</font>

;<br/>
<b>reconsider </b><font color="Maroon" title="c4">BBB</font> = <font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">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> <b>by </b><i><a class="ref" href="sin_cos.html#T61" target="_self" title="SIN_COS:th.61">Th61</a></i>;<br/>
<a NAME="E5:103"/>
<font color="Maroon" title="c4">BBB</font> is <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a>
 
;<br/>
<b>hence </b><a NAME="E6:103"/>
<font color="Maroon" title="c3">z</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K30" title="SIN_COS:func.30">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#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="sin_cos.html#K6" title="SIN_COS:func.6">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="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">z</font>
 <b>by </b><i><a class="txt" href="sin_cos.html#E3:103"><i><font color="Green" title="E87">A2</font></i></a>, <a class="ref" href="comseq_3.html#T57" title="COMSEQ_3:th.57">COMSEQ_3:57</a></i>;<br/>


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