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

<span class="kw">let </span><font color="Maroon" title="c3">p</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</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="Red">::  thesis: </font></span></a><span class="hide">  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E1:80"/><span class="lab"><font color="Green" title="E60">A1</font></span>: 
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"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <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> ] <span class="kw">means </span><span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<span class="lab"><font color="Green" title="E61">A2</font></span>: <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></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 class="p1">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">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 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/>.= 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D4" title="COMSEQ_3:def.4">COMSEQ_3:def 4</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T4" target="_self" title="SIN_COS:th.4">Th4</a>, <a class="ref" href="complex1.html#T15" title="COMPLEX1:th.15">COMPLEX1:15</a></span>
;<br/>
<a NAME="E2:80_1"/><span class="lab"><font color="Green" title="E62">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80_1"><span class="lab"><font color="Green" title="E61">A2</font></span></a></span>;<br/>
<a NAME="E3:80_1"/><span class="lab"><font color="Green" title="E63">A4</font></span>: 
for <font color="Olive" title="b1">k</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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">k</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:80_1_2"/><span class="lab"><font color="Green" title="E64">A5</font></span>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="lab"><font color="Green" title="E65">A6</font></span>: <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span></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">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<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"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <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="#E1:80_1_2"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></span>
<br/>.= 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <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="ref" href="comseq_3.html#D4" title="COMSEQ_3:def.4">COMSEQ_3:def 4</a></span>
<br/>.= 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</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">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!c</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#D8" target="_self" title="SIN_COS:def.8">Def8</a></span>
<br/>.= 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">!</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T37" target="_self" title="SIN_COS:th.37">Th37</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T28" title="COMPLEX1:th.28">COMPLEX1:28</a></span>
;<br/>
<span class="kw">thus </span><a NAME="E3:80_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80_1_2"><span class="lab"><font color="Green" title="E65">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:80_1"/><span class="lab"><font color="Green" title="E64">A7</font></span>: 
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  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E2:80_1"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="txt" href="#E3:80_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>);<br/></span>
<span class="kw">thus </span><a NAME="E5:80_1"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80_1"><span class="lab"><font color="Green" title="E64">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:80"/><span class="lab"><font color="Green" title="E61">A8</font></span>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></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"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">n</font>, <font color="Maroon" title="c5">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">m</font></span><br/>

<a NAME="E1:80_2"/><span class="lab"><font color="Green" title="E62">A9</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E2:80_2"/><span class="lab"><font color="Green" title="E63">A10</font></span>: 
( <font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="txt" href="#E1:80_2"><span class="lab"><font color="Green" title="E62">A9</font></span></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:80_2"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="txt" href="#E2:80_2"><span class="lab"><font color="Green" title="E63">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="lab"><font color="Green" title="E62">A11</font></span>:  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80"><span class="lab"><font color="Green" title="E61">A8</font></span></a>, <a class="ref" href="seq_4.html#T41" title="SEQ_4:th.41">SEQ_4:41</a>, <a class="ref" href="valued_0.html#T24" title="VALUED_0:th.24">VALUED_0:24</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80"><span class="lab"><font color="Green" title="E60">A1</font></span></a></span>
;<br/>
<span class="lab"><font color="Green" title="E63">A12</font></span>:  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> = 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#T54" title="COMSEQ_3:th.54">COMSEQ_3:54</a></span>
<br/>.= 
 <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K4" title="COMSEQ_3:func.4">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T28" title="COMPLEX1:th.28">COMPLEX1:28</a></span>
;<br/>
<span class="kw">thus </span><a NAME="E5:80"/>
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:80"><span class="lab"><font color="Green" title="E62">A11</font></span></a>, <a class="txt" href="#E4:80"><span class="lab"><font color="Green" title="E63">A12</font></span></a>, <a class="ref" href="series_1.html#D3" title="SERIES_1:def.3">SERIES_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
