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

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="csspace.html#NM1" title="CSSPACE:NM.1">ComplexUnitarySpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">seq</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">z</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>  st <font color="Olive" title="b1">seq</font> is <a href="clvect_3.html#V2" title="CLVECT_3:attr.2">absolutely_summable</a> holds <br/><font color="Olive" title="b2">z</font> <a href="clvect_1.html#K7" title="CLVECT_1:func.7">*</a> <font color="Olive" title="b1">seq</font> is <a href="clvect_3.html#V2" title="CLVECT_3:attr.2">absolutely_summable</a></span><br/><b>let </b><font color="Maroon" title="c2">seq</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">z</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>  st <font color="Maroon" title="c2">seq</font> is <a href="clvect_3.html#V2" title="CLVECT_3:attr.2">absolutely_summable</a> holds <br/><font color="Olive" title="b1">z</font> <a href="clvect_1.html#K7" title="CLVECT_1:func.7">*</a> <font color="Maroon" title="c2">seq</font> is <a href="clvect_3.html#V2" title="CLVECT_3:attr.2">absolutely_summable</a></span><br/><b>let </b><font color="Maroon" title="c3">z</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">seq</font> is <a href="clvect_3.html#V2" title="CLVECT_3:attr.2">absolutely_summable</a> implies <font color="Maroon" title="c3">z</font> <a href="clvect_1.html#K7" title="CLVECT_1:func.7">*</a> <font color="Maroon" title="c2">seq</font> is <a href="clvect_3.html#V2" title="CLVECT_3:attr.2">absolutely_summable</a> )</span><br/>





<b>assume </b><a NAME="E1:30"/>
<font color="Maroon" title="c2">seq</font> is <a href="clvect_3.html#V2" title="CLVECT_3:attr.2">absolutely_summable</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">z</font> <a href="clvect_1.html#K7" title="CLVECT_1:func.7">*</a> <font color="Maroon" title="c2">seq</font> is <a href="clvect_3.html#V2" title="CLVECT_3:attr.2">absolutely_summable</a></span><br/>

<a NAME="E2:30"/><b>then </b>
<span class="p1"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 
<b>by </b><i><a class="ref" href="clvect_3.html#D8" target="_self" title="CLVECT_3:def.8">Def8</a></i>;<br/>
<a NAME="E3:30"/><b>then </b><i><font color="Green" title="E14">A1</font></i>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</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#T13" title="SERIES_1:th.13">SERIES_1:13</a></i>;<br/>
<a NAME="E4:30"/>
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="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clvect_1.html#K7" title="CLVECT_1:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</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; <span class="p1"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clvect_1.html#K7" title="CLVECT_1:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</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">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clvect_1.html#K7" title="CLVECT_1:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">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; <span class="p1"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clvect_1.html#K7" title="CLVECT_1:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> )</span><br/>

<b>thus </b><a NAME="E1:30_1"/>
<span class="p1"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clvect_1.html#K7" title="CLVECT_1:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">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> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clvect_1.html#K7" title="CLVECT_1:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:30_1_1"/>
<span class="p1"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clvect_1.html#K7" title="CLVECT_1:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clvect_1.html#K7" title="CLVECT_1:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span>
 
<b>by </b><i><a class="ref" href="clvect_2.html#D3" title="CLVECT_2:def.3">CLVECT_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E2:30_1_1"/>
<span class="p1"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clvect_1.html#K7" title="CLVECT_1:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">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> 
 <b>by </b><i><a class="ref" href="csspace.html#T46" title="CSSPACE:th.46">CSSPACE:46</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:30_1"/>
<span class="p1"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clvect_1.html#K7" title="CLVECT_1:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2"><b>proof </b></a><div class="add">

<span class="p1"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clvect_1.html#K7" title="CLVECT_1:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> = 
<span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clvect_1.html#K7" title="CLVECT_1:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_2.html#D3" title="CLVECT_2:def.3">CLVECT_2:def 3</a></i>
<br/>.= 
<span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#D15" title="CLVECT_1:def.15">CLVECT_1:def 15</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span>
<b>by </b><i><a class="ref" href="csspace.html#T45" title="CSSPACE:th.45">CSSPACE:45</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_2.html#D3" title="CLVECT_2:def.3">CLVECT_2:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
;<br/>
<b>hence </b><a NAME="E2:30_1_2"/>
<span class="p1"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clvect_1.html#K7" title="CLVECT_1:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c3">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E5:30"/><b>then </b>
<span class="p1"><a href="clvect_2.html#K2" title="CLVECT_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clvect_1.html#K7" title="CLVECT_1:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span></span><a href="clvect_2.html#K2" title="CLVECT_2:func.2">.||</a></span> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 
<b>by </b><i><a class="txt" href="#E3:30"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="series_1.html#T24" title="SERIES_1:th.24">SERIES_1:24</a></i>;<br/>
<b>hence </b><a NAME="E6:30"/>
<font color="Maroon" title="c3">z</font> <a href="clvect_1.html#K7" title="CLVECT_1:func.7">*</a> <font color="Maroon" title="c2">seq</font> is <a href="clvect_3.html#V2" title="CLVECT_3:attr.2">absolutely_summable</a>
 <b>by </b><i><a class="ref" href="clvect_3.html#D8" target="_self" title="CLVECT_3:def.8">Def8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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