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<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="E1:58"/><i><font color="Green">E38</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#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> ) &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>2</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#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><span class="p2"><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> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2"><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> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 ;<br/>

<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:58_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="clopban3.html#E1:58"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E2:58_1"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="clvect_1.html#D10">CLVECT_1:def 10</a></i>;<br/><b>hence </b><a NAME="E3:58_1"/>
<span class="p1"><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> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="clvect_1.html#D16">CLVECT_1:def 16</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:58"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><span class="p1">(<span class="default"><span class="p2"><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> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2"><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> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="clopban3.html#E1:58"><i><font color="Green">E38</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2"><b>now </b></a><div class="add"><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="E1:58_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:58_2"/><i><font color="Green">E42</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><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> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><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> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="clopban3.html#T2" target="_self">Th2</a></i>;<br/><i><font color="Green">E43</font></i>: <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2"><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> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2"><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> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>
<br/>.= 
<span class="p1"><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> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="clopban3.html#E2:58_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p2"><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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2"><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> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></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/>.= 
<span class="p1"><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> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="clopban3.html#E2:58_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><b>hence </b><a NAME="E5:58_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3"><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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <span class="p3"><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="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="clopban3.html#E4:58"><i><font color="Green">E40</font></i></a>, <a class="txt" href="clopban3.html#E1:58_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="clopban3.html#E3:58_2"><i><font color="Green">E43</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:58"/>
not <span class="p1"><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> is <a href="series_1.html#V1">summable</a>
 <b>by </b><i><a class="txt" href="clopban3.html#E2:58"><i><font color="Green">E39</font></i></a>, <a class="ref" href="series_1.html#T44">SERIES_1:44</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="clopban3.html#D4" target="_self">CLOPBAN3:def 4</a><br/>


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