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

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

<b>assume </b><a NAME="E1:17"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<a NAME="E2:17"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="lp_space.html#E1:17"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E3:17"/><i><font color="Green">E22</font></i>: 
1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="lp_space.html#E2:17"><i><font color="Green">E21</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E4:17"/><i><font color="Green">E23</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="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E5:17"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_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/>

<a NAME="E1:17_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="lp_space.html#E4:17"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E2:17_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></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_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="lp_space.html#E2:17"><i><font color="Green">E21</font></i></a>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:17_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="lp_space.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:17"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">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">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 
<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>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</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">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:17_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>;<br/>
<a NAME="E2:17_2"/><i><font color="Green">E27</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_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:17_2_1"/><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</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">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_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>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lp_space.html#E5:17"><i><font color="Green">E24</font></i></a></i>
<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">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="lp_space.html#E1:17_2_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E3:17_2"/>
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="lp_space.html#E1:17_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="lp_space.html#E2:17_2"><i><font color="Green">E27</font></i></a>);<br/></i>
<b>hence </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">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>

<br/>.= 
0
<b>by </b><i><a class="txt" href="lp_space.html#E5:17"><i><font color="Green">E24</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E7:17"/>
(  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font> is <a href="series_1.html#V1">summable</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3"><b>proof </b></a><div class="add">

<a NAME="E1:17_3"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>3</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"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></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="E1:17_3_1"/><i><font color="Green">E27</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>take </b>
0
;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E2:17_3_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<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"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</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">0 <a href="real_1.html#K5">-</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="lp_space.html#E6:17"><i><font color="Green">E25</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E4:17_3_1"/>
 <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"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</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="lp_space.html#E1:17_3_1"><i><font color="Green">E27</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:17_3"/><b>then </b><i><font color="Green">E27</font></i>: 
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a></i>;<br/>
<a NAME="E3:17_3"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="ref" href="series_1.html#D2">SERIES_1:def 2</a></i>;<br/>
<a NAME="E4:17_3"/>
 <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">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="lp_space.html#E1:17_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="lp_space.html#E2:17_3"><i><font color="Green">E27</font></i></a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E5:17_3"/>
(  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</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="lp_space.html#E3:17_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="series_1.html#D3">SERIES_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:17"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font> is <a href="series_1.html#V1">summable</a> &amp; <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lp_space.html#K1" target="_self">rto_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="lp_space.html#E3:17"><i><font color="Green">E22</font></i></a>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>;<br/>


</div>
