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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b><span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> $1</span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> $1</span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>;<br/>
<b>consider </b><font color="Maroon" title="c1">IT</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:25_1_1"/><i><font color="Green" title="E24">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">IT</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="csspace.html#K1" title="CSSPACE:func.1">the_set_of_ComplexSequences</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:25_1_1"/><i><font color="Green" title="E25">A2</font></i>: 
not <font color="Maroon" title="c1">IT</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:25_1_1_1"/>
<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <a href="csspace.html#K6" title="CSSPACE:func.6">CZeroseq</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <a href="csspace.html#K6" title="CSSPACE:func.6">CZeroseq</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c2">rseq</font> = <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <a href="csspace.html#K6" title="CSSPACE:func.6">CZeroseq</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <a href="csspace.html#K6" title="CSSPACE:func.6">CZeroseq</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>thus </b><font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <a href="csspace.html#K6" title="CSSPACE:func.6">CZeroseq</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <a href="csspace.html#K6" title="CSSPACE:func.6">CZeroseq</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <a href="csspace.html#K6" title="CSSPACE:func.6">CZeroseq</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <a href="csspace.html#K6" title="CSSPACE:func.6">CZeroseq</a> </span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14" title="COMSEQ_1:def.14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <a href="csspace.html#K6" title="CSSPACE:func.6">CZeroseq</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 0
<b>by </b><i><a class="ref" href="csspace.html#D6" target="_self" title="CSSPACE:def.6">Def6</a>, <a class="ref" href="complex1.html#T130" title="COMPLEX1:th.130">COMPLEX1:130</a></i>
<br/>.= 
0

;<br/></div><b>end;</b></div>
<a NAME="E3:25_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">rseq</font> is <a href="series_1.html#V2" title="SERIES_1:attr.2">absolutely_summable</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#T3" title="COMSEQ_3:th.3">COMSEQ_3:3</a></i>;<br/>
<b>hence </b><a NAME="E4:25_1_1_1_1"/>
<span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <a href="csspace.html#K6" title="CSSPACE:func.6">CZeroseq</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <a href="csspace.html#K6" title="CSSPACE:func.6">CZeroseq</a> </span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 <b>by </b><i><a class="ref" href="series_1.html#T40" title="SERIES_1:th.40">SERIES_1:40</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:25_1_1_1"/>
not <font color="Maroon" title="c1">IT</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="csspace.html#E2:25_1_1"><i><font color="Green" title="E24">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:25_1_1"/>
<font color="Maroon" title="c1">IT</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="csspace.html#K1" title="CSSPACE:func.1">the_set_of_ComplexSequences</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:25_1_1_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">IT</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="csspace.html#K1" title="CSSPACE:func.1">the_set_of_ComplexSequences</a> 
 
<b>by </b><i><a class="txt" href="csspace.html#E2:25_1_1"><i><font color="Green" title="E24">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:25_1_1_2"/>
<font color="Maroon" title="c1">IT</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="csspace.html#K1" title="CSSPACE:func.1">the_set_of_ComplexSequences</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:25_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="csspace.html#K7" title="CSSPACE:func.7">Linear_Space_of_ComplexSequences</a>  st <br/>( not <font color="Olive">b<sub>1</sub></font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> iff ( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="csspace.html#K1" title="CSSPACE:func.1">the_set_of_ComplexSequences</a>  &amp; <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">x</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">x</font></span>)</span></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> ) ) ) )
 <b>by </b><i><a class="txt" href="csspace.html#E2:25_1_1"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="csspace.html#E3:25_1_1"><i><font color="Green" title="E25">A2</font></i></a></i>;<br/>


</div>
