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

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:116"/><i><font color="Green">E73</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>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a> 
 ;<br/>

<a NAME="E2:116"/><i><font color="Green">E74</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="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</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 <font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:116_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="comseq_3.html#D7">COMSEQ_3:def 7</a></i>;<br/>
<a NAME="E2:116_1"/><i><font color="Green">E76</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="116_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:116_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<a href="complex1.html#K5">0c</a>  <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#D6">COMPLEX1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="csspace.html#E1:116"><i><font color="Green">E73</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="csspace.html#E1:116_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="comseq_3.html#D7">COMSEQ_3:def 7</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E3:116_1"/>
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="csspace.html#E1:116_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="csspace.html#E2:116_1"><i><font color="Green">E76</font></i></a>);<br/></i>
<b>hence </b><span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>

<br/>.= 
 <a href="complex1.html#K5">0c</a> 
<b>by </b><i><a class="txt" href="csspace.html#E1:116"><i><font color="Green">E73</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E3:116"/>
(  <a href="comseq_3.html#K8">Sum</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a>  &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="comseq_3.html#V1">summable</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_2"><b>proof </b></a><div class="add">

<a NAME="E1:116_2"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</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/><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">c<sub>4</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <a href="complex1.html#K5">0c</a> </span>)</span></span><a href="complex1.html#K17">.|</a></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="116_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:116_2_1"/><i><font color="Green">E76</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

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

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E2:116_2_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>thus </b><a NAME="E3:116_2_1"/>
<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">c<sub>4</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="complex1.html#K11">-</a> <a href="complex1.html#K5">0c</a> </span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="csspace.html#E2:116"><i><font color="Green">E74</font></i></a>, <a class="txt" href="csspace.html#E1:116_2_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="complex1.html#T130">COMPLEX1:130</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:116_2"/><b>then </b><i><font color="Green">E76</font></i>: 
 <a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">c<sub>4</sub></font> is <a href="comseq_2.html#V2">convergent</a>
 
<b>by </b><i><a class="ref" href="comseq_2.html#D4">COMSEQ_2:def 4</a></i>;<br/>
<a NAME="E3:116_2"/><b>then </b>
 <a href="comseq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7">Partial_Sums</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a> 
 
<b>by </b><i><a class="txt" href="csspace.html#E1:116_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="comseq_2.html#D5">COMSEQ_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:116_2"/>
(  <a href="comseq_3.html#K8">Sum</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a>  &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="comseq_3.html#V1">summable</a> )
 <b>by </b><i><a class="txt" href="csspace.html#E2:116_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="comseq_3.html#D8">COMSEQ_3:def 8</a>, <a class="ref" href="comseq_3.html#D10">COMSEQ_3:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:116"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="comseq_3.html#V1">summable</a> &amp;  <a href="comseq_3.html#K8">Sum</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a>  )
 ;<br/>


</div>
