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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <a href="sin_cos.html#K3" target="_self">Prod_real_n</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="newton.html#K12">!</a> ;<br/>
<a NAME="E1:8_1_1"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="sin_cos.html#D5" target="_self">Def5</a>, <a class="ref" href="newton.html#T18">NEWTON:18</a></i>;<br/>
<div><i><font color="Green">E5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1"><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="E1:8_1_1_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 ;<br/><a href="sin_cos.html#K3" target="_self">Prod_real_n</a>  <a href="seq_1.html#K2">.</a> <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">(<span class="default"><a href="sin_cos.html#K3" target="_self">Prod_real_n</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> 
<b>by </b><i><a class="txt" href="sin_cos.html#E1:8_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="newton.html#T21">NEWTON:21</a></i>
;<br/><b>hence </b><a NAME="E3:8_1_1_1"/>
<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>
<a NAME="E3:8_1_1"/>
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#E1:8_1_1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="sin_cos.html#E2:8_1_1"><i><font color="Green">E5</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:8_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K5">!</a>  iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K3" target="_self">Prod_real_n</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>


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