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

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

<b>assume </b><a NAME="E1:10"/>
<font color="Maroon">f</font> is <a href="series_1.html#V2">absolutely_summable</a>
 ;<br/>

<a NAME="E2:10"/><b>then </b>
(  <a href="partfun3.html#K7">abs</a> <font color="Maroon">f</font> is <a href="series_1.html#V1">summable</a> &amp; <font color="Maroon">f</font> is <a href="series_1.html#V1">summable</a> )
 
<b>by </b><i><a class="ref" href="series_1.html#D5">SERIES_1:def 5</a>, <a class="ref" href="series_1.html#T40">SERIES_1:40</a></i>;<br/>
<a NAME="E3:10"/><b>then </b><i><font color="Green">E37</font></i>: 
(  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><a href="partfun3.html#K7">abs</a> <font color="Maroon">f</font></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">f</font> is <a href="seq_2.html#V4">convergent</a> )
 
<b>by </b><i><a class="ref" href="series_1.html#D2">SERIES_1:def 2</a></i>;<br/>
<a NAME="E4:10"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="partfun3.html#K7">abs</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">f</font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T26">SEQ_4:26</a></i>;<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">(<span class="default"><a href="partfun3.html#K7">abs</a> <span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="partfun3.html#K7">abs</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<span class="p1">(<span class="default"><a href="partfun3.html#K7">abs</a> <span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0 = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> 0</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"><a href="partfun3.html#K7">abs</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>
;<br/>
<a NAME="E6:10"/><b>then </b><i><font color="Green">E39</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/>
<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>now </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="E1:10_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/><a NAME="E2:10_2"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="partfun3.html#K7">abs</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>;<br/><a NAME="E3:10_2"/><b>then </b>
( <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">f</font></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="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.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></span><a href="complex1.html#K17">.|</a></span> &amp; <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.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></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <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"><a href="partfun3.html#K7">abs</a> <font color="Maroon">f</font></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"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.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></span><a href="complex1.html#K17">.|</a></span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="complex1.html#T142">COMPLEX1:142</a></i>;<br/><a NAME="E4:10_2"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">f</font></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="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <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"><a href="partfun3.html#K7">abs</a> <font color="Maroon">f</font></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"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.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></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:10_2"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.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></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <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"><a href="partfun3.html#K7">abs</a> <font color="Maroon">f</font></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"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.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></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>;<br/><a NAME="E6:10_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="partfun3.html#K7">abs</a> <span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">f</font></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> <a href="xxreal_0.html#R1">&lt;=</a> <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"><a href="partfun3.html#K7">abs</a> <font color="Maroon">f</font></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"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.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></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>;<br/><a NAME="E7:10_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="partfun3.html#K7">abs</a> <span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">f</font></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> <a href="xxreal_0.html#R1">&lt;=</a> <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"><a href="partfun3.html#K7">abs</a> <font color="Maroon">f</font></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"><a href="partfun3.html#K7">abs</a> <font color="Maroon">f</font></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="seq_1.html#T16">SEQ_1:16</a></i>;<br/><b>hence </b><a NAME="E8:10_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="series_1.html#D1">SERIES_1:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:10"/>
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>
<a NAME="E9:10"/><b>then </b>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="partfun3.html#K7">abs</a> <span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <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"><a href="partfun3.html#K7">abs</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="seq_2.html#T32">SEQ_2:32</a></i>;<br/>
<a NAME="E10:10"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <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">Partial_Sums</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <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"><a href="partfun3.html#K7">abs</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="seq_4.html#T27">SEQ_4:27</a></i>;<br/>
<a NAME="E11:10"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <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">Partial_Sums</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="partfun3.html#K7">abs</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="series_1.html#D3">SERIES_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E12:10"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="partfun3.html#K7">abs</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="series_1.html#D3">SERIES_1:def 3</a></i>;<br/>


</div>
