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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:29"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:29"/><i><font color="Green">E28</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default">1,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="newton.html#K10">In_Power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#D4">NEWTON:def 4</a></i>;<br/>
<div><i><font color="Green">E29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:29_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E2:29_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E3:29_1"/>
<span class="p1">(<span class="default"><a href="irrat_1.html#K3" target="_self">cseq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="newton.html#K10">In_Power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="irrat_1.html#E1:29"><i><font color="Green">E27</font></i></a>, <a class="ref" href="irrat_1.html#T21" target="_self">Th21</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:29_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E2:29_2"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="irrat_1.html#K3" target="_self">cseq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K9">choose</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="irrat_1.html#NK1" target="_self">.^.</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="irrat_1.html#D3" target="_self">Def3</a></i>
<br/>.= 
0 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="irrat_1.html#NK1" target="_self">.^.</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="irrat_1.html#E2:29_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="newton.html#D3">NEWTON:def 3</a></i>
<br/>.= 
0

;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:29"/>
 <a href="irrat_1.html#K3" target="_self">cseq</a> <font color="Maroon">c<sub>1</sub></font> is <a href="series_1.html#V1">summable</a>
 <b>by </b><i><a class="txt" href="irrat_1.html#E2:29"><i><font color="Green">E28</font></i></a>, <a class="txt" href="irrat_1.html#E3:29"><i><font color="Green">E29</font></i></a>, <a class="ref" href="irrat_1.html#T19" target="_self">Th19</a></i>;<br/>

<b>thus </b><span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="irrat_1.html#NK1" target="_self">.^.</a> <font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="power.html#T48">POWER:48</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default">1,<span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="newton.html#K10">In_Power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T41">NEWTON:41</a></i>
<br/>.= 
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="irrat_1.html#K3" target="_self">cseq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="irrat_1.html#E2:29"><i><font color="Green">E28</font></i></a>, <a class="txt" href="irrat_1.html#E3:29"><i><font color="Green">E29</font></i></a>, <a class="txt" href="irrat_1.html#E4:29"><i><font color="Green">E30</font></i></a>, <a class="ref" href="irrat_1.html#T19" target="_self">Th19</a></i>
;<br/>

<b>hence </b><a NAME="E7:29"/>
<a href="irrat_1.html#K4" target="_self">dseq</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="irrat_1.html#K3" target="_self">cseq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="irrat_1.html#D4" target="_self">Def4</a></i>;<br/>


</div>
