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<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span> = 
 <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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p4">(<span class="default">0 <a href="nat_1.html#K1">+</a> 0</span>)</span></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="sin_cos.html#T49" target="_self">Th49</a></i>
<br/>.= 
 <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">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K8">+</a> <span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</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#E82"><i><font color="Green">Lemma62</font></i></a></i>
<br/>.= 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,0</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="complex1.html#K9">*</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,0</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>
<b>by </b><i><a class="txt" href="sin_cos.html#E42"><i><font color="Green">Lemma36</font></i></a></i>
<br/>.= 
 <a href="complex1.html#K3">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">Re</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,0</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="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p5">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p0">(<span class="default"><span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,0</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></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,0</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>
<b>by </b><i><a class="ref" href="complex1.html#T29">COMPLEX1:29</a></i>
<br/>.= 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,0</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="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,0</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> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p3">(<span class="default"><span class="p4"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,0</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>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
<br/>.= 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,0</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="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,0</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> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,0</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="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p5">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p0">(<span class="default"><span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,0</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></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T29">COMPLEX1:29</a></i>
<br/>.= 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,0</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="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,0</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> <a href="complex1.html#K9">*</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,0</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="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,0</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></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
<br/>.= 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,0</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> <a href="complex1.html#K9">*</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,0</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="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,0</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></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T49" target="_self">Th49</a></i>
<br/>.= 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,0</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="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,0</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></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T46" target="_self">Th46</a></i>
<br/>.= 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p4">(<span class="default"><a href="comseq_3.html#K8">Sum</a> <span class="p5">(<span class="default"><span class="p0"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,0</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></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T49" target="_self">Th49</a></i>
<br/>.= 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span></span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T46" target="_self">Th46</a></i>
<br/>.= 
 <a href="complex1.html#K3">Re</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default">0 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="sin_cos.html#K7" 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="txt" href="sin_cos.html#E68"><i><font color="Green">Lemma51</font></i></a></i>
<br/>.= 
 <a href="complex1.html#K3">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#K2">Sum</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">0 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="sin_cos.html#K7" 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></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T28">COMPLEX1:28</a></i>
;<br/>
<b>hence </b><a NAME="E3:89"/>
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="sin_cos.html#K7" 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"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span></span>)</span>
 ;<br/>


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