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

<b>let </b><font color="Maroon">th</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

 <a href="comseq_3.html#K8">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#T54">COMSEQ_3:54</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
;<br/>
<a NAME="E2:73"/><b>then </b>
 <a href="comseq_3.html#K8">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>;<br/>
<a NAME="E3:73"/><b>then </b>
(  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="complex1.html#T28">COMPLEX1:28</a></i>;<br/>
<a NAME="E4:73"/><b>then </b><i><font color="Green">E53</font></i>: 
(  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> &amp;  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="series_1.html#D2">SERIES_1:def 2</a>, <a class="ref" href="series_1.html#D3">SERIES_1:def 3</a></i>;<br/>
<div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:73_2"/>
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:73_2"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/><b>take </b><font color="Maroon">n</font> = <font color="Maroon">n</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:73_2_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:73_2_1"/><i><font color="Green">E59</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sin_cos.html#E3:73_2"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E3:73_2_1"/>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E4:73_2_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="sin_cos.html#E7:73"><i><font color="Green">E61</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>hence </b><a NAME="E5:73_2_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="txt" href="sin_cos.html#E8:73"><i><font color="Green">E62</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:73_2"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E6:73"/><b>then </b>
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a></i>;<br/>
<a NAME="E7:73"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<div><i><font color="Green">E62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:73_3"/>
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:73_3"/><i><font color="Green">E91</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/><b>take </b><font color="Maroon">n</font> = <font color="Maroon">n</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:73_3_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:73_3_1"/><i><font color="Green">E93</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:73_3_1"/>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E4:73_3_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E5:73_3_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E6:73_3_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="sin_cos.html#E8:73"><i><font color="Green">E62</font></i></a>, <a class="ref" href="sin_cos.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:73_3"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E9:73"/><b>then </b>
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a></i>;<br/>
<a NAME="E10:73"/><b>then </b>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E7:73"><i><font color="Green">E61</font></i></a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E11:73"/>
(  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K25" target="_self">P_sin</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> &amp;  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K26" target="_self">P_cos</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">th</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="sin_cos.html#K6" target="_self">ExpSeq</a> </span>)</span></span>)</span> )
 <b>by </b><i>, , <a class="txt" href="sin_cos.html#E7:73"><i><font color="Green">E61</font></i></a>, <a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a>, <a class="ref" href="series_1.html#D3">SERIES_1:def 3</a></i>;<br/>


</div>
