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

<b>let </b><font color="Maroon">th</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:98"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">th</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:98"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1"><b>proof </b></a><div class="add">

<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><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1;<br/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0 = 
<span class="p1">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">th</font> <a href="newton.html#K2">|^</a> 0</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">0 <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
1
<b>by </b><i>, <a class="ref" href="newton.html#T18">NEWTON:18</a></i>
;<br/>
<a NAME="E2:98_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E3:98_1"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_2"><b>proof </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="E1:98_1_2"/><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<i><font color="Green">E61</font></i>: <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">th</font> <a href="newton.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E3:98_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">th</font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="sin_cos.html#D15" target="_self">Def15</a>, <a class="ref" href="prepower.html#T13">PREPOWER:13</a></i>;<br/>
<a NAME="E4:98_1_2"/>
<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#K5" target="_self">!</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E5:98_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">th</font> <a href="newton.html#K2">|^</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="xcmplx_0.html#K7">/</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="sin_cos.html#K5" target="_self">!</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E6:98_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">th</font> <a href="newton.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="sin_cos.html#K5" target="_self">!</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<b>hence </b><a NAME="E7:98_1_2"/>
<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="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:98_1"/>
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>(<a class="txt" href="sin_cos.html#E21"><i><font color="Green">Lemma82</font></i></a>, );<br/></i>
<b>hence </b><a NAME="E5:98_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:98"/>
<font color="Maroon">th</font> is   <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="E4:98"/><b>then </b>
<font color="Maroon">th</font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a>  is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i/>;<br/>
<a NAME="E5:98"/><b>then </b>
(  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K7" target="_self">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"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="series_1.html#D2">SERIES_1:def 2</a>, <a class="ref" href="series_1.html#D3">SERIES_1:def 3</a></i>;<br/>
<a NAME="E6:98"/><b>then </b>
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">th</font> <a href="sin_cos.html#K7" target="_self">ExpSeq</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="sin_cos.html#D17" target="_self">Def17</a>, </i>;<br/>
<b>hence </b><a NAME="E7:98"/>
<a href="sin_cos.html#K27" target="_self">exp_R</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="sin_cos.html#T3" target="_self">Th3</a></i>;<br/>


</div>
