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<div class="add">

<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="E1:13"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 ;<br/>

<a NAME="E2:13"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prepower.html#K3">|^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><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>3</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> = 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D9">SIN_COS:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></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="seq_1.html#T13">SEQ_1:13</a></i>
;<br/>
<b>hence </b><a NAME="E2:13_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><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>3</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:13"/>
<font color="Maroon">c<sub>2</sub></font> <a href="sin_cos.html#K7">ExpSeq</a>  is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T49">SIN_COS:49</a></i>;<br/>
<a NAME="E4:13"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="sin_cos.html#K7">ExpSeq</a>  is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <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="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="series_1.html#T7">SERIES_1:7</a></i>;<br/>
<a NAME="E5:13"/><b>then </b><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> 0 )
 
<b>by </b><i><a class="ref" href="seq_2.html#T21">SEQ_2:21</a>, <a class="ref" href="seq_2.html#T22">SEQ_2:22</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E6:13"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:13"/><i><font color="Green">E14</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="taylor_2.html#E6:13"><i><font color="Green">E13</font></i></a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a>, <a class="txt" href="taylor_2.html#E5:13"><i><font color="Green">E12</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:13_2"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:13_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="power.html#T3">POWER:3</a>, <a class="txt" href="taylor_2.html#E1:13"><i><font color="Green">E10</font></i></a></i>;<br/><a NAME="E3:13_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="sin_cos.html#K5">!</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="newton.html#T23">NEWTON:23</a></i>;<br/><a NAME="E4:13_2"/><i><font color="Green">E18</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="taylor_2.html#E2:13_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="taylor_2.html#E3:13_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="real_2.html#T125">REAL_2:125</a></i>;<br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K14">(#)</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sin_cos.html#K7">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <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="prepower.html#K3">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_2.html#E2:13"><i><font color="Green">E11</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prepower.html#K3">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="taylor_2.html#E1:13"><i><font color="Green">E10</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><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>5</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a>, <a class="txt" href="taylor_2.html#E4:13_2"><i><font color="Green">E18</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <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>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
;<br/><b>hence </b><a NAME="E6:13_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <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>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="taylor_2.html#E8:13"><i><font color="Green">E14</font></i></a>, <a class="txt" href="taylor_2.html#E1:13_2"><i><font color="Green">E15</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:13"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <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="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="sin_cos.html#K5">!</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div>
