<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E2:5"/><i><font color="Green">E5</font></i>: 
 <a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="sin_cos.html#T60">SIN_COS:60</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ] means <span class="p1">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prepower.html#K12">#R</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K28">exp_R</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<a NAME="E3:5"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="taylor_1.html#E2:5"><i><font color="Green">E5</font></i></a>, <a class="ref" href="prepower.html#T85">PREPOWER:85</a>, <a class="ref" href="sin_cos.html#T56">SIN_COS:56</a></i>;<br/>
<a NAME="E4:5"/><i><font color="Green">E7</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</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="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:5_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prepower.html#K12">#R</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prepower.html#K12">#R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prepower.html#K13">#R</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E2:5"><i><font color="Green">E5</font></i></a>, <a class="ref" href="prepower.html#T89">PREPOWER:89</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prepower.html#K12">#R</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K29">exp_R</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E2:5"><i><font color="Green">E5</font></i></a>, <a class="ref" href="prepower.html#T86">PREPOWER:86</a></i>
<br/>.= 
 <a href="sin_cos.html#K29">exp_R</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E1:5_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="sin_cos.html#T55">SIN_COS:55</a></i>
<br/>.= 
 <a href="sin_cos.html#K28">exp_R</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

;<br/>


</div><b>end;</b></div>
<a NAME="E5:5"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</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#S2">NAT_1:sch 2</a>(<a class="txt" href="taylor_1.html#E3:5"><i><font color="Green">E6</font></i></a>, <a class="txt" href="taylor_1.html#E4:5"><i><font color="Green">E7</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:5"/>
<span class="p1">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="prepower.html#K12">#R</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K28">exp_R</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>


</div>
