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

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span><span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</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="complex1.html#K5" title="COMPLEX1:func.5">0c</a> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></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> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1;<br/>
<span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</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="complex1.html#K5" title="COMPLEX1:func.5">0c</a> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></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"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></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/>.= 
<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="complex1.html#K5" title="COMPLEX1:func.5">0c</a> <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></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T18" title="VALUED_1:th.18">VALUED_1:18</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T3" target="_self" title="SIN_COS:th.3">Th4</a>, <a class="ref" href="complex1.html#T48" title="COMPLEX1:th.48">COMPLEX1:48</a></span>
;<br/>
<a NAME="E2:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E3:26"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:26_2"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
<span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</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="complex1.html#K5" title="COMPLEX1:func.5">0c</a> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:26_2"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
<font color="Maroon" title="c1">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#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</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="complex1.html#K5" title="COMPLEX1:func.5">0c</a> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></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="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
1 <a href="real_1.html#K7" title="REAL_1:func.7">+</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="complex1.html#K5" title="COMPLEX1:func.5">0c</a> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_2"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></span>
<br/>.= 
1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <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="complex1.html#K5" title="COMPLEX1:func.5">0c</a> <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="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T18" title="VALUED_1:th.18">VALUED_1:18</a></span>
<br/>.= 
1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> <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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<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>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_2"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="sin_cos.html#T3" target="_self" title="SIN_COS:th.3">Th4</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T44" title="COMPLEX1:th.44">COMPLEX1:44</a></span>
;<br/> <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 NAME="E4:26"/>
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>);<br/></span>
<a NAME="E5:26"/><span class="kw">then </span>
 <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</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="complex1.html#K5" title="COMPLEX1:func.5">0c</a> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_0.html#D18" title="VALUED_0:def.18">VALUED_0:def 18</a></span>;<br/>
<a NAME="E6:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A5</font></span>: 
<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="complex1.html#K5" title="COMPLEX1:func.5">0c</a> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></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 class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#D2" title="SERIES_1:def.2">SERIES_1:def 2</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span><span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1;<br/>
<span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> <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 class="p1">(<span class="default"><a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> <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 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/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T3" target="_self" title="SIN_COS:th.3">Th4</a></span>
;<br/>
<a NAME="E8:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A6</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E9:26"/><span class="lab"><font color="Green" title="E26">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>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">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"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c1">n</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:26_4"/><span class="lab"><font color="Green" title="E27">A8</font></span>: 
<span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> <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> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> <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="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> <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="p2">(<span class="default"><font color="Maroon" title="c1">n</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="#E1:26_4"><span class="lab"><font color="Green" title="E27">A8</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#K6" title="COMPLEX1:func.6">1r</a> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</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"><a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> <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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a></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="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<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>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T3" target="_self" title="SIN_COS:th.3">Th4</a></span>
<br/>.= 
1

; <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 NAME="E10:26"/>
 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>2</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="#E8:26"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E26">A7</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E11:26"/>
( <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a>  is  <a href="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a>  &amp;  <a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p1">(<span class="default"><a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="ref" href="comseq_2.html#T10" title="COMSEQ_2:th.10">COMSEQ_2:10</a>, <a class="ref" href="comseq_3.html#D9" title="COMSEQ_3:def.9">COMSEQ_3:def 9</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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