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

<b>let </b><font color="Maroon">seq</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/><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="E1:5"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><a href="comseq_3.html#K4">Im</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> 0 )
 ;<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"><span class="p2">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></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">seq</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="E2:5"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> 0 = 
<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="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> 0</span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> 0</span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_3.html#D7">COMSEQ_3:def 7</a></i>
<br/>.= 
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">seq</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>
;<br/>
<b>hence </b><a NAME="E2:5_1"/>
<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><b>end;</b></div>
<div><i><font color="Green">E35</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>now </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>assume </b><a NAME="E1:5_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font>]
 ;<br/><b>thus </b><a NAME="E2:5_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:5_2_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">i</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">(<span class="default"><a href="comseq_3.html#K3">Re</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</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>2</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="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K3">Re</a> <font color="Maroon">seq</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#NR2" target="_self">&gt;=</a> 0;<br/>
<a NAME="E1:5_2_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:5_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K3">Re</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> 0
 
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:5_2_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[0]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_1_1_2"><b>now </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="E1:5_2_1_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>]
 ;<br/><b>thus </b><a NAME="E2:5_2_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:5_2_1_1_2_1"/><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K3">Re</a> <font color="Maroon">seq</font></span>)</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> <a href="hidden.html#R1">=</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="comseq_3.html#K3">Re</a> <font color="Maroon">seq</font></span>)</span></span>)</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">(<span class="default"><a href="comseq_3.html#K3">Re</a> <font color="Maroon">seq</font></span>)</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><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>;<br/>
<a NAME="E2:5_2_1_1_2_1"/>
( <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K3">Re</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <font color="Maroon">seq</font></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> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:5_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K3">Re</a> <font color="Maroon">seq</font></span>)</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> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E4:5_2_1_1_2_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E3:5_2_1_1"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E4:5_2_1_1"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K3">Re</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:5_2_1"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">i</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">(<span class="default"><a href="comseq_3.html#K4">Im</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</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>2</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="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K4">Im</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0;<br/>
<a NAME="E1:5_2_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:5_2_1_2_1"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K4">Im</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K4">Im</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> 0
 
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:5_2_1_2_1"/>
<font color="Maroon">S<sub>2</sub></font>[0]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E60</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_1_2_2"><b>now </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="E1:5_2_1_2_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>]
 ;<br/><b>thus </b><a NAME="E2:5_2_1_2_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_1_2_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:5_2_1_2_2_1"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K4">Im</a> <font color="Maroon">seq</font></span>)</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> <a href="hidden.html#R1">=</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="comseq_3.html#K4">Im</a> <font color="Maroon">seq</font></span>)</span></span>)</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">(<span class="default"><a href="comseq_3.html#K4">Im</a> <font color="Maroon">seq</font></span>)</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><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:5_2_1_2_2_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E3:5_2_1_2"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E4:5_2_1_2"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K4">Im</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>


</div><b>end;</b></div>
<i><font color="Green">E62</font></i>: <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></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">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<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="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">m</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="comseq_1.html#D14">COMSEQ_1:def 14</a></i>
<br/>.= 
<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="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_3.html#D7">COMSEQ_3:def 7</a></i>
;<br/>
<a NAME="E4:5_2_1"/>
( <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K3">Re</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K4">Im</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:5_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="comseq_3.html#T26">COMSEQ_3:26</a></i>;<br/>
<a NAME="E6:5_2_1"/><b>then </b><i><font color="Green">E63</font></i>: 
(  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="comseq_3.html#D3">COMSEQ_3:def 3</a>, <a class="ref" href="comseq_3.html#D4">COMSEQ_3:def 4</a></i>;<br/>
<a NAME="E7:5_2_1"/>
( <span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><a href="comseq_3.html#K4">Im</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i/>;<br/>
<a NAME="E8:5_2_1"/><b>then </b><i><font color="Green">E64</font></i>: 
(  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="comseq_3.html#D3">COMSEQ_3:def 3</a>, <a class="ref" href="comseq_3.html#D4">COMSEQ_3:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">z1</font> = <span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font>;<br/>
<b>set </b><font color="Maroon">z2</font> = <font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<a NAME="E9:5_2_1"/><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:5_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, , <a class="ref" href="real_2.html#T121">REAL_2:121</a></i>;<br/>
<b>then </b><span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></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">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<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="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">m</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">seq</font> <a href="comseq_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">m</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>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</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">seq</font> <a href="comseq_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">m</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="comseq_1.html#D14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">seq</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#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">seq</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">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>
;<br/>
<b>hence </b><a NAME="E12:5_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</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></div><b>end;</b></div>
<a NAME="E4:5"/>
for <font color="Olive">m</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">m</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E5:5"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">seq</font></span>)</span></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> <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">seq</font></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 ;<br/>


</div>
