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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>,<font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>;<br/>
<span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>,0 = 
1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">0 <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="complfld.html#T10">COMPLFLD:10</a>, <a class="ref" href="group_1.html#D8">GROUP_1:def 8</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default">0 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default">0 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
<b>by </b><i><a class="ref" href="sin_cos.html#T34">SIN_COS:34</a></i>
;<br/>
<a NAME="E2:62"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><i><font color="Green">E60</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:62_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 ;<br/><b>then </b><span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span> <a href="group_1.html#K10">*</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
<b>by </b><i><a class="ref" href="group_1.html#D8">GROUP_1:def 8</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
<b>by </b><i><a class="ref" href="hahnban1.html#T11">HAHNBAN1:11</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
<b>by </b><i><a class="ref" href="sin_cos.html#T80">SIN_COS:80</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
<b>by </b><i><a class="ref" href="sin_cos.html#T80">SIN_COS:80</a></i>
;<br/><b>hence </b><a NAME="E3:62_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E4:62"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="comptrig.html#E2:62"><i><font color="Green">E59</font></i></a>, <a class="txt" href="comptrig.html#E3:62"><i><font color="Green">E60</font></i></a>);</i><br/>


</div>
