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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:27"/><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 <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>take </b>
0
; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a>,<a class="ref" href="series_1.html#D2">SERIES_1:def 2</a>,<a class="ref" href="clopban3.html#D4" target="_self">CLOPBAN3:def 4</a><br/>

<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="E2:27"/><i><font color="Green">E26</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="E3:27"/>
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"><a href="clvect_1.html#K10">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_1.html#K10">.||</a></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="clopban3.html#E1:27"><i><font color="Green">E25</font></i></a>, <a class="ref" href="clopban3.html#T6" target="_self">Th6</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="E5:27"/>
 <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="clvect_1.html#K10">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_1.html#K10">.||</a></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="clopban3.html#E2:27"><i><font color="Green">E26</font></i></a></i>;<br/>


</div>
