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<span class="kw">let </span><font color="Maroon" title="c3">rseq1</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_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">seq</font> being  <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>  st <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Olive" title="b1">seq</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is  <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a>  &amp; ( for <font color="Olive" title="b2">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  <font color="Maroon" title="c3">rseq1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Olive" title="b1">seq</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default">2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Olive" title="b2">n</font></span>)</span></span>)</span> ) holds <br/>( <font color="Olive" title="b1">seq</font> is  <a href="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a>  iff <font color="Maroon" title="c3">rseq1</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">seq</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"> ( <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">seq</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is  <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a>  &amp; ( 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  <font color="Maroon" title="c3">rseq1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">seq</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default">2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> ) implies ( <font color="Maroon" title="c4">seq</font> is  <a href="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a>  iff <font color="Maroon" title="c3">rseq1</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:98"/>
( <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">seq</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is  <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a>  &amp; ( 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  <font color="Maroon" title="c3">rseq1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">seq</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default">2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> ) )
 ; <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="c4">seq</font> is  <a href="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a>  iff <font color="Maroon" title="c3">rseq1</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  )</span><br/>

<a NAME="E2:98"/><span class="kw">then </span>
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/> ( <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">seq</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is  <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a>  &amp; <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">seq</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#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">rseq1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">seq</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default">2 <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="comseq_3.html#E14"><span class="lab"><font color="Green" title="E5">Lm3</font></span></a></span>;<br/>
<a NAME="E3:98"/><span class="kw">then </span>
( <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c4">seq</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>  iff <font color="Maroon" title="c3">rseq1</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#T35" title="SERIES_1:th.35">SERIES_1:35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:98"/>
( <font color="Maroon" title="c4">seq</font> is  <a href="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a>  iff <font color="Maroon" title="c3">rseq1</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D11" target="_self" title="COMSEQ_3:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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