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<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<a NAME="E1:20"/><i><font color="Green">E31</font></i>: 
 <a href="sin_cos.html#K19">sin</a>  <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T34">FDIFF_1:34</a>, <a class="ref" href="sin_cos.html#T73">SIN_COS:73</a></i>;<br/>
<a NAME="E2:20"/><i><font color="Green">E32</font></i>: 
 <a href="sin_cos.html#K22">cos</a>  <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T34">FDIFF_1:34</a>, <a class="ref" href="sin_cos.html#T72">SIN_COS:72</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <span class="p1">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> );<br/>
<a NAME="E3:20"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<i><font color="Green">E41</font></i>: <span class="p1">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> = 
<a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="ref" href="taylor_1.html#D5">TAYLOR_1:def 5</a></i>
<br/>.= 
1 <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T33">RFUNCT_1:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> 0</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>
;<br/>
<i><font color="Green">E42</font></i>: <span class="p1">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="ref" href="taylor_1.html#D5">TAYLOR_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="ref" href="taylor_1.html#D5">TAYLOR_1:def 5</a></i>
<br/>.= 
<a href="sin_cos.html#K19">sin</a>  <a href="fdiff_1.html#K2">`|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="ref" href="fdiff_2.html#T16">FDIFF_2:16</a>, </i>
<br/>.= 
<a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i/>
<br/>.= 
1 <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T33">RFUNCT_1:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> 0</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>
;<br/>
<i><font color="Green">E48</font></i>: <span class="p1">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> = 
<a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="ref" href="taylor_1.html#D5">TAYLOR_1:def 5</a></i>
<br/>.= 
1 <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T33">RFUNCT_1:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> 0</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>
;<br/>
<span class="p1">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="ref" href="taylor_1.html#D5">TAYLOR_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="ref" href="taylor_1.html#D5">TAYLOR_1:def 5</a></i>
<br/>.= 
<a href="sin_cos.html#K22">cos</a>  <a href="fdiff_1.html#K2">`|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="ref" href="fdiff_2.html#T16">FDIFF_2:16</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_1.html#K16">-</a> <a href="sin_cos.html#K19">sin</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i/>
<br/>.= 
1 <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <a href="sin_cos.html#K19">sin</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T33">RFUNCT_1:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> 0</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <a href="sin_cos.html#K19">sin</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> 0</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos.html#K19">sin</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T38">RFUNCT_1:38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> 0</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T65">RFUNCT_1:65</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> 0</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T29">RFUNCT_1:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
;<br/>
<b>hence </b><a NAME="E5:20_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:20"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:20_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<a NAME="E2:20_2"/><i><font color="Green">E58</font></i>: 
( <a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; <a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span> )
 
<b>by </b><i><a class="ref" href="fdiff_2.html#T16">FDIFF_2:16</a>, , </i>;<br/>
<i><font color="Green">E59</font></i>: <span class="p1">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="ref" href="taylor_1.html#D5">TAYLOR_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="fdiff_2.html#T19">FDIFF_2:19</a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="fdiff_2.html#T16">FDIFF_2:16</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <a href="sin_cos.html#K19">sin</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos.html#K19">sin</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T38">RFUNCT_1:38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T65">RFUNCT_1:65</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T29">RFUNCT_1:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
;<br/>
<i><font color="Green">E60</font></i>: <span class="p1">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="ref" href="taylor_1.html#D5">TAYLOR_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="fdiff_2.html#T19">FDIFF_2:19</a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="fdiff_2.html#T16">FDIFF_2:16</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i/>
;<br/>
<i><font color="Green">E61</font></i>: <span class="p1">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="ref" href="taylor_1.html#D5">TAYLOR_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="fdiff_2.html#T19">FDIFF_2:19</a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="fdiff_2.html#T16">FDIFF_2:16</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i/>
;<br/>
<span class="p1">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="ref" href="taylor_1.html#D5">TAYLOR_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="fdiff_2.html#T19">FDIFF_2:19</a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="fdiff_2.html#T16">FDIFF_2:16</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K16">-</a> <a href="sin_cos.html#K19">sin</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <a href="sin_cos.html#K19">sin</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T38">RFUNCT_1:38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T65">RFUNCT_1:65</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T29">RFUNCT_1:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
;<br/>
<b>hence </b><a NAME="E7:20_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:20"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">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="E6:20"/>
( <span class="p1">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22">cos</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="taylor_1.html#K5">diff</a> <a href="sin_cos.html#K22">cos</a> ,<span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="prepower.html#K3">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> )
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