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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">a<sub>1</sub></font> <a href="sin_cos.html#K4" target="_self">!c</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span>;<br/>
<a NAME="E1:71"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="sin_cos.html#T1" target="_self">Th1</a>, <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a>, <a class="ref" href="newton.html#T18">NEWTON:18</a></i>;<br/>
<a NAME="E2:71"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:71_1"/><i><font color="Green">E58</font></i>: 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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="complex1.html#T27">COMPLEX1:27</a></i>;<br/>
<b>assume </b><a NAME="E2:71_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="sin_cos.html#K4" target="_self">!c</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span>
 ;<br/>

<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a>  = 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="sin_cos.html#T1" target="_self">Th1</a>, <a class="txt" href="sin_cos.html#E1:71_1"><i><font color="Green">E58</font></i></a></i>
<br/>.= 
<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"><font color="Maroon">c<sub>1</sub></font> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default">0 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</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/>.= 
<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"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span>,0</span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="newton.html#T21">NEWTON:21</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:71"/>
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="sin_cos.html#E1:71"><i><font color="Green">E56</font></i></a>, <a class="txt" href="sin_cos.html#E2:71"><i><font color="Green">E57</font></i></a>);</i><br/>


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