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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:17"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="series_1.html#V1" target="_self">summable</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means  <a href="series_1.html#K2" target="_self">Sum</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E2:17"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = <a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span> as   <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i><a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
<a NAME="E2:17_1"/><i><font color="Green">E18</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>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> 0
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E3:17_1"/><i><font color="Green">E19</font></i>: 
 <a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="txt" href="series_1.html#E1:17"><i><font color="Green">E16</font></i></a>, <a class="ref" href="series_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E4:17_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seqm_3.html#K1">^\</a> 1 is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T33">SEQ_4:33</a></i>;<br/>
<a NAME="E5:17_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T39">SEQ_4:39</a></i>;<br/>
<a NAME="E6:17_1"/>
 <a href="series_1.html#K1" target="_self">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="series_1.html#E2:17_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="series_1.html#T14" target="_self">Th14</a></i>;<br/>
<b>then </b> <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="series_1.html#E4:17_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="series_1.html#E5:17_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="seq_2.html#T26">SEQ_2:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="series_1.html#E3:17_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="seq_4.html#T33">SEQ_4:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K2" target="_self">Sum</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K2" target="_self">Sum</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="seq_4.html#T41">SEQ_4:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K2" target="_self">Sum</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="series_1.html#E2:17_1"><i><font color="Green">E18</font></i></a></i>
;<br/>
<a NAME="E8:17_1"/><b>then </b>
 <a href="series_1.html#K2" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" target="_self">Sum</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span>
 
;<br/>
<a NAME="E9:17_1"/><b>then </b>
 <a href="series_1.html#K2" target="_self">Sum</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E10:17_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i><a class="ref" href="series_1.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:17"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</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="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>proof </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="E1:17_2"/><i><font color="Green">E19</font></i>: 
 <a href="series_1.html#K2" target="_self">Sum</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> as   <a href="seq_1.html#NM1">Real_Sequence</a> <b>by </b><i><a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
<a NAME="E3:17_2"/><i><font color="Green">E20</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>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 0
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E4:17_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T39">SEQ_4:39</a></i>;<br/>
<a NAME="E5:17_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K1">^\</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> is <a href="series_1.html#V1" target="_self">summable</a>
 
<b>by </b><i><a class="txt" href="series_1.html#E1:17"><i><font color="Green">E16</font></i></a>, <a class="ref" href="series_1.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E6:17_2"/><b>then </b><i><font color="Green">E22</font></i>: 
 <a href="series_1.html#K1" target="_self">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="series_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E7:17_2"/><b>then </b><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="seqm_3.html#K1">^\</a> 1 is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T33">SEQ_4:33</a></i>;<br/>
<a NAME="E8:17_2"/><i><font color="Green">E24</font></i>: 
 <a href="series_1.html#K1" target="_self">Partial_Sums</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K1">^\</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K10">-</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="series_1.html#E3:17_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="series_1.html#T14" target="_self">Th14</a></i>;<br/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> = 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K1">^\</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#T36">SEQM_3:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K1">^\</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="series_1.html#E4:17_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="series_1.html#E7:17_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="series_1.html#E8:17_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="seq_2.html#T26">SEQ_2:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K1">^\</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="series_1.html#E6:17_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="seq_4.html#T33">SEQ_4:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K2" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K2" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="seq_4.html#T41">SEQ_4:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K2" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="series_1.html#E3:17_2"><i><font color="Green">E20</font></i></a></i>
;<br/>
<b>then </b> <a href="series_1.html#K2" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="series_1.html#K2" target="_self">Sum</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K1">^\</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="series_1.html#E1:17_2"><i><font color="Green">E19</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K2" target="_self">Sum</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K2" target="_self">Sum</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></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" target="_self">Def1</a></i>
;<br/>
<b>hence </b><a NAME="E11:17_2"/>
 <a href="series_1.html#K2" target="_self">Sum</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:17"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</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#S1">NAT_1:sch 1</a>(<a class="txt" href="series_1.html#E2:17"><i><font color="Green">E17</font></i></a>, <a class="txt" href="series_1.html#E3:17"><i><font color="Green">E18</font></i></a>);</i><br/>


</div>
