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<div class="add">

<b>let </b><font color="Maroon" title="c1">th</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

 <a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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#K4" title="COMSEQ_3:func.4">Im</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">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>
<b>by </b><i><a class="ref" href="comseq_3.html#T54" title="COMSEQ_3:th.54">COMSEQ_3:54</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><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"><span class="p5"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span>,<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#K4" title="COMSEQ_3:func.4">Im</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T27" title="COMPLEX1:th.27">COMPLEX1:27</a></i>
;<br/>
<a NAME="E2:71"/><b>then </b>
 <a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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#K4" title="COMSEQ_3:func.4">Im</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">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>
 
<b>by </b><i><a class="ref" href="complex1.html#T27" title="COMPLEX1:th.27">COMPLEX1:27</a></i>;<br/>
<a NAME="E3:71"/><b>then </b>
(  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</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"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span> &amp;  <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"><span class="p3"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</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"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="complex1.html#T28" title="COMPLEX1:th.28">COMPLEX1:28</a></i>;<br/>
<a NAME="E4:71"/><b>then </b><i><font color="Green" title="E57">A1</font></i>: 
(  <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <span class="p2">(<span class="default"><span class="p3"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <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#K3" title="COMSEQ_3:func.3">Re</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</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"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span> &amp;  <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</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"><span class="p3"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <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"><span class="p4"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</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"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="series_1.html#D2" title="SERIES_1:def.2">SERIES_1:def 2</a>, <a class="ref" href="series_1.html#D3" title="SERIES_1:def.3">SERIES_1:def 3</a></i>;<br/>
<div><i><font color="Green" title="E58">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">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> ;<br/><b>assume </b><a NAME="E1:71_2"/>
<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><b>then consider </b><font color="Maroon" title="c3">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> <b> such that </b><br/><a NAME="E3:71_2"/><i><font color="Green" title="E59">A3</font></i>: 
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" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">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" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="txt" href="sin_cos.html#E4:71"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></i>;<br/><b>take </b><font color="Maroon" title="c4">n</font> = <font color="Maroon" title="c3">n</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2_1"><b>now </b></a><div class="add"><b>let </b><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> ;<br/><b>assume </b><a NAME="E1:71_2_1"/><i><font color="Green" title="E60">A4</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 ;<br/><a NAME="E2:71_2_1"/><i><font color="Green" title="E61">A5</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">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" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K26" title="SIN_COS:func.26">P_cos</a> </span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">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" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K3" title="COMSEQ_3:func.3">Re</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sin_cos.html#T38" target="_self" title="SIN_COS:th.38">Th38</a></i>;<br/><a NAME="E3:71_2_1"/>
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">k</font>
 ;<br/><a NAME="E4:71_2_1"/><b>then </b>
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="txt" href="sin_cos.html#E1:71_2_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><b>hence </b><a NAME="E5:71_2_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">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" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K26" title="SIN_COS:func.26">P_cos</a> </span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="txt" href="sin_cos.html#E3:71_2"><i><font color="Green" title="E59">A3</font></i></a>, <a class="txt" href="sin_cos.html#E2:71_2_1"><i><font color="Green" title="E61">A5</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:71_2"/>
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" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">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" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K26" title="SIN_COS:func.26">P_cos</a> </span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">p</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E6:71"/><b>then </b>
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K26" title="SIN_COS:func.26">P_cos</a> </span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#D6" title="SEQ_2:def.6">SEQ_2:def 6</a></i>;<br/>
<a NAME="E7:71"/><b>then </b><i><font color="Green" title="E59">A6</font></i>: 
 <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"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K26" title="SIN_COS:func.26">P_cos</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</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"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E5:71"><i><font color="Green" title="E58">A2</font></i></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></i>;<br/>
<div><i><font color="Green" title="E60">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">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> ;<br/><b>assume </b><a NAME="E1:71_3"/>
<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><b>then consider </b><font color="Maroon" title="c3">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> <b> such that </b><br/><a NAME="E3:71_3"/><i><font color="Green" title="E61">A8</font></i>: 
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" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">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" title="SERIES_1:func.1">Partial_Sums</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"><span class="p0"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="txt" href="sin_cos.html#E4:71"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></i>;<br/><b>take </b><font color="Maroon" title="c4">n</font> = <font color="Maroon" title="c3">n</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_3_1"><b>now </b></a><div class="add"><b>let </b><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> ;<br/><b>assume </b><a NAME="E1:71_3_1"/><i><font color="Green" title="E62">A9</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 ;<br/><a NAME="E2:71_3_1"/><i><font color="Green" title="E63">A10</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">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" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K25" title="SIN_COS:func.25">P_sin</a> </span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">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" title="SERIES_1:func.1">Partial_Sums</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"><span class="p0"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sin_cos.html#T38" target="_self" title="SIN_COS:th.38">Th38</a></i>;<br/><a NAME="E3:71_3_1"/>
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">k</font>
 ;<br/><a NAME="E4:71_3_1"/><b>then </b>
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="txt" href="sin_cos.html#E1:71_3_1"><i><font color="Green" title="E62">A9</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E5:71_3_1"/><b>then </b>
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E6:71_3_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">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" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K25" title="SIN_COS:func.25">P_sin</a> </span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="txt" href="sin_cos.html#E3:71_3"><i><font color="Green" title="E61">A8</font></i></a>, <a class="txt" href="sin_cos.html#E2:71_3_1"><i><font color="Green" title="E63">A10</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:71_3"/>
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" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">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" title="SERIES_1:func.1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K25" title="SIN_COS:func.25">P_sin</a> </span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p4">(<span class="default"><span class="p5"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">p</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E9:71"/><b>then </b>
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K25" title="SIN_COS:func.25">P_sin</a> </span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#D6" title="SEQ_2:def.6">SEQ_2:def 6</a></i>;<br/>
<a NAME="E10:71"/><b>then </b>
 <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"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K25" title="SIN_COS:func.25">P_sin</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</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"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E8:71"><i><font color="Green" title="E60">A7</font></i></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E11:71"/>
(  <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K25" title="SIN_COS:func.25">P_sin</a> </span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K25" title="SIN_COS:func.25">P_sin</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</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"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span> &amp;  <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K26" title="SIN_COS:func.26">P_cos</a> </span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="sin_cos.html#K26" title="SIN_COS:func.26">P_cos</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</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"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">th</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="sin_cos.html#K6" title="SIN_COS:func.6">ExpSeq</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="sin_cos.html#E5:71"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="sin_cos.html#E7:71"><i><font color="Green" title="E59">A6</font></i></a>, <a class="txt" href="sin_cos.html#E8:71"><i><font color="Green" title="E60">A7</font></i></a>, <a class="ref" href="seq_2.html#D6" title="SEQ_2:def.6">SEQ_2:def 6</a>, <a class="ref" href="series_1.html#D3" title="SERIES_1:def.3">SERIES_1:def 3</a></i>;<br/>


</div>
