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

<b>let </b><font color="Maroon">cseq</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:10"/><i><font color="Green">E30</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  <font color="Maroon">cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a> 
 ;<br/>

<a NAME="E2:10"/><i><font color="Green">E33</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"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">cseq</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><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/>

<a NAME="E1:10_1"/>
<font color="Maroon">cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:10_1"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">cseq</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T130">COMPLEX1:130</a>, <a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:10"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">m</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"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">cseq</font></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<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"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">cseq</font></span><a href="comseq_1.html#K9">.|</a></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"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">cseq</font></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:10_2"/><i><font color="Green">E35</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="E2:10_2"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">k</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">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:10_2_1"/><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">cseq</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</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"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">cseq</font></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 ;<br/>

<b>thus </b><span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">cseq</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</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"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">cseq</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">cseq</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">cseq</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">cseq</font></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E3:10_2"/>
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>(, );<br/></i>
<b>hence </b><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">cseq</font></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> = 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">cseq</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>

<br/>.= 
0
<b>by </b><i/>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E4:10"/>
(  <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">cseq</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">cseq</font> is <a href="comseq_3.html#V2">absolutely_summable</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3"><b>proof </b></a><div class="add">

<a NAME="E1:10_3"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">p</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#NR3" target="_self">&lt;</a> <font color="Olive">p</font> holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</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"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">cseq</font></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</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="E1:10_3_1"/><i><font color="Green">E53</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 ;<br/>

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

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E2:10_3_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</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"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">cseq</font></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</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/>
<br/>.= 
0
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E4:10_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"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">cseq</font></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:10_3"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">cseq</font></span><a href="comseq_1.html#K9">.|</a></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="E3:10_3"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">cseq</font></span><a href="comseq_1.html#K9">.|</a></span> 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="E4:10_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"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">cseq</font></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E5:10_3"/>
(  <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">cseq</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">cseq</font> is <a href="comseq_3.html#V2">absolutely_summable</a> )
 <b>by </b><i>, <a class="ref" href="comseq_3.html#D11">COMSEQ_3:def 11</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="E5:10"/>
( <font color="Maroon">cseq</font> is <a href="comseq_3.html#V2">absolutely_summable</a> &amp;  <a href="series_1.html#K2">Sum</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">cseq</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>


</div>
