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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:17"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:17"/>
 <a href="prepower.html#K1">abs</a> <span class="p1">(<span class="default">0 <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>;<br/>
<a NAME="E3:17"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default">0 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default">0 <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="txt" href="taylor_2.html#E1:17"><i><font color="Green">E12</font></i></a>, <a class="ref" href="rcomp_1.html#T8">RCOMP_1:8</a></i>;<br/>
<a NAME="E4:17"/><b>then </b><i><font color="Green">E13</font></i>: 
0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="taylor_2.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E5:17"/>
 <a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K27">exp_R</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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="sin_cos.html#K7">ExpSeq</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:17_1_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="taylor_2.html#E1:17_1_1"><i><font color="Green">E14</font></i></a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K27">exp_R</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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="taylor_1.html#K5">Taylor</a> <a href="sin_cos.html#K27">exp_R</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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>,0,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="taylor_1.html#K4">diff</a> <a href="sin_cos.html#K27">exp_R</a> ,<span class="p5"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#D6">TAYLOR_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K2">|</a> <span class="p4"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="taylor_2.html#T6" target="_self">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span>
<b>by </b><i><a class="txt" href="taylor_2.html#E4:17"><i><font color="Green">E13</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="sin_cos.html#D9">SIN_COS:def 9</a>, <a class="ref" href="sin_cos2.html#T13">SIN_COS2:13</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:17_1"/>
 <a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K27">exp_R</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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="sin_cos.html#K7">ExpSeq</a> 
 <b>by </b><i><a class="ref" href="seq_1.html#T8">SEQ_1:8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:17"/>
(  <a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K27">exp_R</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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="sin_cos.html#K7">ExpSeq</a>  &amp;  <a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K27">exp_R</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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>,<font color="Maroon">c<sub>2</sub></font> is <a href="series_1.html#V2">absolutely_summable</a> &amp; <a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="taylor_2.html#K1" target="_self">Maclaurin</a> <a href="sin_cos.html#K27">exp_R</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">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="taylor_2.html#T15" target="_self">Th15</a>, <a class="ref" href="sin_cos.html#D26">SIN_COS:def 26</a></i>;<br/>


</div>
