<?xml version="1.0"?>
<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>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><font color="Maroon">a<sub>1</sub></font> <a href="sin_cos.html#K4" target="_self">!c</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0;<br/>
<b>thus </b><a NAME="E1:11"/>
0 <a href="sin_cos.html#K4" target="_self">!c</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i/>;<br/>

<a NAME="E2:11"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_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>assume </b><a NAME="E1:11_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/><a NAME="E2:11_1"/><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E3:11_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, , <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:11"/>
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>(, );<br/></i>
<b>hence </b><a NAME="E5:11"/>
<font color="Maroon">n</font> <a href="sin_cos.html#K4" target="_self">!c</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>thus </b><a NAME="E6:11"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K4" target="_self">!c</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="sin_cos.html#K4" target="_self">!c</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/>


</div>
