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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">x</font></span>)</span></span><a href="arytm_0.html#K5">*]</a></span> <a href="newton.html#K2">|^</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="polyeq_3.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><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">x</font></span>)</span></span>)</span> <a href="polyeq_3.html#K1" target="_self">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>;<br/>
<a NAME="E1:43"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="sin_cos.html#T34">SIN_COS:34</a>, <a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/>
<div><i><font color="Green">E47</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>set </b><font color="Maroon">a'</font> =  <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"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</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">n</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>;<br/><b>set </b><font color="Maroon">b'</font> =  <a href="complex1.html#K4">Im</a> <span class="p1"><a href="arytm_0.html#K5">[*</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">n</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</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">n</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>;<br/><b>set </b><font color="Maroon">c'</font> =  <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"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">x</font></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>;<br/><b>set </b><font color="Maroon">d'</font> =  <a href="complex1.html#K4">Im</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">x</font></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>;<br/><a NAME="E1:43_1"/><i><font color="Green">E48</font></i>: 
( <span class="p1"><a href="arytm_0.html#K5">[*</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">n</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</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">n</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="polyeq_3.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="polyeq_3.html#K1" target="_self">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> &amp; <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">x</font></span>)</span></span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">x</font></span>)</span> <a href="polyeq_3.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">x</font></span>)</span> <a href="polyeq_3.html#K1" target="_self">*</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/><b>assume </b><a NAME="E2:43_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/><b>then </b><span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">x</font></span>)</span></span><a href="arytm_0.html#K5">*]</a></span> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<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">n</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="polyeq_3.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="polyeq_3.html#K1" target="_self">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">x</font></span>)</span> <a href="polyeq_3.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">x</font></span>)</span> <a href="polyeq_3.html#K1" target="_self">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="polyeq_3.html#K2" target="_self">+</a> <span class="p1">(<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">n</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</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">x</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">x</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">n</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polyeq_3.html#K1" target="_self">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="polyeq_3.html#K2" target="_self">+</a> <span class="p1">(<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">n</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</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">x</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">x</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">n</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polyeq_3.html#K1" target="_self">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T80">SIN_COS:80</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="polyeq_3.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="polyeq_3.html#K1" target="_self">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T80">SIN_COS:80</a></i>
;<br/><b>hence </b><a NAME="E4:43_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E3:43"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, <a class="txt" href="polyeq_3.html#E1"><i><font color="Green">Lemma19</font></i></a>);</i><br/>


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