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

<span class="kw">let </span><font color="Maroon" title="c3">seq1</font>, <font color="Maroon" title="c4">seq2</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">n</font> being   <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>  holds <br/> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c3">seq1</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> &amp; <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c3">seq1</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">seq2</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> ) ) &amp; <font color="Maroon" title="c4">seq2</font> is  <a href="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a>  implies ( <font color="Maroon" title="c3">seq1</font> is  <a href="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a>  &amp;  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c3">seq1</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">seq2</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:94"/>
( ( for <font color="Olive" title="b1">n</font> being   <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>  holds <br/> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c3">seq1</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> &amp; <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c3">seq1</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">seq2</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> ) ) &amp; <font color="Maroon" title="c4">seq2</font> is  <a href="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">seq1</font> is  <a href="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a>  &amp;  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c3">seq1</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">seq2</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> )</span><br/>

<span class="kw">hence </span><a NAME="E2:94"/>
( <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c3">seq1</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  &amp;  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c3">seq1</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">seq2</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#T24" title="SERIES_1:th.24">SERIES_1:24</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="comseq_3.html#D11" target="_self" title="COMSEQ_3:def.11">COMSEQ_3:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
