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

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="lopban_2.html#NM3" title="LOPBAN_2:NM.3">Banach_Algebra</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">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Olive" title="b1">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 holds <br/>( <font color="Olive" title="b1">z</font> <a href="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</a>  is  <a href="lopban_3.html#V1" title="LOPBAN_3:attr.1">summable</a>  &amp; <font color="Olive" title="b1">z</font> <a href="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</a>  is  <a href="lopban_3.html#V2" title="LOPBAN_3:attr.2">norm_summable</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>; <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="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 implies ( <font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</a>  is  <a href="lopban_3.html#V1" title="LOPBAN_3:attr.1">summable</a>  &amp; <font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</a>  is  <a href="lopban_3.html#V2" title="LOPBAN_3:attr.2">norm_summable</a>  ) )</span><br/>



<a NAME="E1:63"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
 <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="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#T8" title="NORMSP_1:th.8">NORMSP_1:8</a></span>;<br/>
<a NAME="E2:63"/><span class="lab"><font color="Green" title="E34">A2</font></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/> (  <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="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</a> </span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</a> </span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <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> ] <span class="kw">means </span>(  <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="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</a> </span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1 &amp; <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</a> </span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1 );<br/>
<a NAME="E1:63_1"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
<span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</a> </span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D10" title="NORMSP_1:def.10">NORMSP_1:def 10</a></span>;<br/>
<a NAME="E2:63_1"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
for <font color="Olive" title="b1">k</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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">k</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<a NAME="E1:63_1_1"/><span class="lab"><font color="Green" title="E37">A5</font></span>: 
 <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="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#T8" title="NORMSP_1:th.8">NORMSP_1:8</a></span>;<br/>
<a NAME="E2:63_1_1"/>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_2.html#D9" title="LOPBAN_2:def.9">LOPBAN_2:def 9</a></span>;<br/>
<a NAME="E3:63_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A6</font></span>: 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</a> </span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D10" title="NORMSP_1:def.10">NORMSP_1:def 10</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:63_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<a NAME="E5:63_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</a> </span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1_1"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/>
<a NAME="E6:63_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A7</font></span>: 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1_1"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</a> </span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D10" title="NORMSP_1:def.10">NORMSP_1:def 10</a></span>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">z</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_3.html#D13" target="_self" title="LOPBAN_3:def.13">Def13</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:63_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:63_1_1"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="ref" href="normsp_1.html#T8" title="NORMSP_1:th.8">NORMSP_1:8</a>, <a class="ref" href="prepower.html#T4" title="PREPOWER:th.4">PREPOWER:4</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/>


</div><span class="kw">end;</span></div>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</a> </span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> = 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">X</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_3.html#D13" target="_self" title="LOPBAN_3:def.13">Def13</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_2.html#D10" title="LOPBAN_2:def.10">LOPBAN_2:def 10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#T4" title="PREPOWER:th.4">PREPOWER:4</a></span>
;<br/>
<a NAME="E4:63_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A8</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="normsp_1.html#T8" title="NORMSP_1:th.8">NORMSP_1:8</a></span>;<br/>
<a NAME="E5:63_1"/>
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">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E4:63_1"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E2:63_1"><span class="lab"><font color="Green" title="E36">A4</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:63_1"/>
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="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</a> </span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</a> </span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> )
 ; <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><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E3:63"/>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <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="c2">z</font> <a href="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</a>  is  <a href="lopban_3.html#V1" title="LOPBAN_3:attr.1">summable</a>  &amp; <font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</a>  is  <a href="lopban_3.html#V2" title="LOPBAN_3:attr.2">norm_summable</a>  )</span><br/>

<a NAME="E4:63"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/>
<a NAME="E5:63"/><span class="kw">then </span>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a>  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#T28" title="SERIES_1:th.28">SERIES_1:28</a></span>;<br/>
<a NAME="E6:63"/><span class="kw">then </span>
<span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</a> </span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="series_1.html#T24" title="SERIES_1:th.24">SERIES_1:24</a></span>;<br/>
<a NAME="E7:63"/><span class="kw">then </span>
<font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</a>  is  <a href="lopban_3.html#V2" title="LOPBAN_3:attr.2">norm_summable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_3.html#D4" target="_self" title="LOPBAN_3:def.4">Def4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:63"/>
( <font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</a>  is  <a href="lopban_3.html#V1" title="LOPBAN_3:attr.1">summable</a>  &amp; <font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K5" title="LOPBAN_3:func.5">GeoSeq</a>  is  <a href="lopban_3.html#V2" title="LOPBAN_3:attr.2">norm_summable</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>
