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<div class="add">

<b>set </b><font color="Maroon">a</font> = 1 <a href="real_1.html#K6">/</a> 2;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="prepower.html#K2">GeoSeq</a>  as   <a href="seq_1.html#NM1">Real_Sequence</a> ;<br/>
<a NAME="E2:37_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E3:37_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="ref" href="series_1.html#T28">SERIES_1:28</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E4:37_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="prepower.html#T4">PREPOWER:4</a></i>;<br/>
<a NAME="E5:37_1_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:37_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<a NAME="E2:37_1_1_1"/>
<font color="Maroon">x</font> <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="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="prepower.html#T4">PREPOWER:4</a></i>;<br/>
<b>hence </b><a NAME="E3:37_1_1_1"/>
<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="real_2.html#T121">REAL_2:121</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:37_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">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="E7:37_1_1"/><b>then </b>
<font color="Maroon">x</font> is <a href="series_1.html#V2">absolutely_summable</a>
 
<b>by </b><i>, <a class="ref" href="series_1.html#T41">SERIES_1:41</a></i>;<br/>
<a NAME="E8:37_1_1"/><b>then </b>
 <a href="rsspace.html#K2">seq_id</a> <font color="Maroon">x</font> is <a href="series_1.html#V2">absolutely_summable</a>
 
<b>by </b><i><a class="ref" href="rsspace.html#T3">RSSPACE:3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">v</font> = <font color="Maroon">x</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="rsspace3.html#K3">l1_Space</a>  <b>by </b><i><a class="ref" href="rsspace3.html#T8">RSSPACE3:8</a></i>;<br/>
<span class="p1">(<span class="default"><a href="rsspace.html#K2">seq_id</a> <font color="Maroon">v</font></span>)</span> <a href="seq_1.html#K2">.</a> 0 = 
<font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="rsspace.html#T3">RSSPACE:3</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="prepower.html#T4">PREPOWER:4</a></i>
;<br/>
<a NAME="E11:37_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rsspace.html#K6">Zeroseq</a> 
 
<b>by </b><i><a class="ref" href="rsspace.html#D6">RSSPACE:def 6</a></i>;<br/>
<b>hence </b><a NAME="E12:37_1_1"/>
not  <a href="rsspace3.html#K3">l1_Space</a>  is <a href="realset2.html#V3">trivial</a>
 <b>by </b><i><a class="ref" href="anproj_1.html#D8">ANPROJ_1:def 8</a>, <a class="ref" href="rsspace3.html#T8">RSSPACE3:8</a></i>;<br/>


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