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<div><span class="kw">theorem </span><a NAME="T72"><span class="comment"><font color="firebrick">:: COMSEQ_3:72</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">rseq1</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><br/>  for <font color="Olive" title="b2">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="b2">seq</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is  <a href="valued_0.html#V8" title="VALUED_0:attr.8">non-increasing</a>  &amp; (  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b1">rseq1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="b2">seq</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default">2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Olive" title="b3">n</font></span>)</span></span>)</span> ) holds <br/>( <font color="Olive" title="b2">seq</font> is  <a href="comseq_3.html#V2" title="COMSEQ_3:attr.2">absolutely_summable</a>  iff <font color="Olive" title="b1">rseq1</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  )</div></div>
