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

<span class="kw">let </span><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</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="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">p</font> = <font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c6">q</font> = <font color="Maroon" title="c4">q</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
 <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c6">q</font></span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span> = 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c6">q</font></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T45" target="_self" title="SIN_COS:th.45">Th49</a></span>
<br/>.= 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="sin_cos.html#E37"><span class="lab"><font color="Green" title="E35">Lm2</font></span></a></span>
<br/>.= 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p5">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T13" title="COMPLEX1:th.13">COMPLEX1:13</a></span>
<br/>.= 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p5">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p5">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">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>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T13" title="COMPLEX1:th.13">COMPLEX1:13</a></span>
<br/>.= 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p5">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p5">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">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>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T45" target="_self" title="SIN_COS:th.45">Th49</a></span>
<br/>.= 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></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> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p5">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">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>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T42" target="_self" title="SIN_COS:th.42">Th46</a></span>
<br/>.= 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p5">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="sin_cos.html#K4" title="SIN_COS:func.4">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>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T45" target="_self" title="SIN_COS:th.45">Th49</a></span>
<br/>.= 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></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#T42" target="_self" title="SIN_COS:th.42">Th46</a></span>
<br/>.= 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</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"><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>

<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T12" title="COMPLEX1:th.12">COMPLEX1:12</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:78"/>
 <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">rExpSeq</a></span>)</span></span>)</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/>


</div>
