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

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="lopban_2.html#NM3" title="LOPBAN_2:NM.3">Banach_Algebra</a>;<br/><b>let </b><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>;<br/>



<b>assume </b><a NAME="E1:73"/><i><font color="Green" title="E43">A1</font></i>: 
<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> 1
 ;<br/>

<a NAME="E2:73"/>
 <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>
 
<b>by </b><i><a class="ref" href="normsp_1.html#T8" title="NORMSP_1:th.8">NORMSP_1:8</a></i>;<br/>
<a NAME="E3:73"/><b>then </b><i><font color="Green" title="E44">A2</font></i>: 
 <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#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="lopban_3.html#E1:73"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E4:73"/><i><font color="Green" title="E45">A3</font></i>: 
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#K8" title="LOPBAN_3:func.8">GeoSeq</a> </span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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#K8" title="LOPBAN_3:func.8">GeoSeq</a> </span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>proof </b></a><div class="add">

<b>defpred </b><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> ] <b>means </b>(  <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#K8" title="LOPBAN_3:func.8">GeoSeq</a> </span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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#K8" title="LOPBAN_3:func.8">GeoSeq</a> </span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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#K2" title="SEQ_1:func.2">.</a> $1 );<br/>
<a NAME="E1:73_1"/><i><font color="Green" title="E46">A4</font></i>: 
<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#K8" title="LOPBAN_3:func.8">GeoSeq</a> </span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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#K8" title="LOPBAN_3:func.8">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_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>
 
<b>by </b><i><a class="ref" href="normsp_1.html#D10" title="NORMSP_1:def.10">NORMSP_1:def 10</a></i>;<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"><font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K8" title="LOPBAN_3:func.8">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_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>
<b>by </b><i><a class="ref" href="lopban_3.html#D13" target="_self" title="LOPBAN_3:def.13">Def13</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="lopban_2.html#D10" title="LOPBAN_2:def.10">LOPBAN_2:def 10</a></i>
<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#K2" title="SEQ_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="prepower.html#T4" title="PREPOWER:th.4">PREPOWER:4</a></i>
;<br/>
<a NAME="E3:73_1"/><b>then </b><i><font color="Green" title="E47">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="lopban_3.html#E1:73_1"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="normsp_1.html#T8" title="NORMSP_1:th.8">NORMSP_1:8</a></i>;<br/>
<a NAME="E4:73_1"/><i><font color="Green" title="E48">A6</font></i>: 
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="73_1_2"><b>proof </b></a><div class="add">

<b>let </b><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> ;<br/>

<b>assume </b><a NAME="E1:73_1_2"/><i><font color="Green" title="E49">A7</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>]
 ;<br/>

<a NAME="E2:73_1_2"/><i><font color="Green" title="E50">A8</font></i>: 
 <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>
 
<b>by </b><i><a class="ref" href="normsp_1.html#T8" title="NORMSP_1:th.8">NORMSP_1:8</a></i>;<br/>
<i><font color="Green" title="E51">A9</font></i>: <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#K8" title="LOPBAN_3:func.8">GeoSeq</a> </span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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#K8" title="LOPBAN_3:func.8">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_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>
<b>by </b><i><a class="ref" href="normsp_1.html#D10" title="NORMSP_1:def.10">NORMSP_1:def 10</a></i>
<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#K8" title="LOPBAN_3:func.8">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_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>
<b>by </b><i><a class="ref" href="lopban_3.html#D13" target="_self" title="LOPBAN_3:def.13">Def13</a></i>
;<br/>
<a NAME="E4:73_1_2"/>
<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#K8" title="LOPBAN_3:func.8">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_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#K8" title="LOPBAN_3:func.8">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_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>
 
<b>by </b><i><a class="ref" href="lopban_2.html#D9" title="LOPBAN_2:def.9">LOPBAN_2:def 9</a></i>;<br/>
<a NAME="E5:73_1_2"/><b>then </b><i><font color="Green" title="E52">A10</font></i>: 
<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#K8" title="LOPBAN_3:func.8">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_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#K8" title="LOPBAN_3:func.8">GeoSeq</a> </span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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>
 
<b>by </b><i><a class="ref" href="normsp_1.html#D10" title="NORMSP_1:def.10">NORMSP_1:def 10</a></i>;<br/>
<a NAME="E6:73_1_2"/>
<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#K8" title="LOPBAN_3:func.8">GeoSeq</a> </span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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#K2" title="SEQ_1:func.2">.</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>
 
<b>by </b><i><a class="txt" href="lopban_3.html#E1:73_1_2"><i><font color="Green" title="E49">A7</font></i></a>, <a class="txt" href="lopban_3.html#E2:73_1_2"><i><font color="Green" title="E50">A8</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<a NAME="E7:73_1_2"/><b>then </b>
<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#K8" title="LOPBAN_3:func.8">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_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#K2" title="SEQ_1:func.2">.</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>
 
<b>by </b><i><a class="txt" href="lopban_3.html#E5:73_1_2"><i><font color="Green" title="E52">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E8:73_1_2"/>
(  <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#K8" title="LOPBAN_3:func.8">GeoSeq</a> </span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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> &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#K8" title="LOPBAN_3:func.8">GeoSeq</a> </span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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> <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#K2" title="SEQ_1:func.2">.</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> )
 <b>by </b><i><a class="txt" href="lopban_3.html#E3:73_1_2"><i><font color="Green" title="E51">A9</font></i></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></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:73_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>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="lopban_3.html#E3:73_1"><i><font color="Green" title="E47">A5</font></i></a>, <a class="txt" href="lopban_3.html#E4:73_1"><i><font color="Green" title="E48">A6</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:73_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#K8" title="LOPBAN_3:func.8">GeoSeq</a> </span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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#K8" title="LOPBAN_3:func.8">GeoSeq</a> </span>)</span></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:73"/>
<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>
 
<b>by </b><i><a class="txt" href="lopban_3.html#E3:73"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="series_1.html#T28" title="SERIES_1:th.28">SERIES_1:28</a></i>;<br/>
<a NAME="E6:73"/><b>then </b>
<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#K8" title="LOPBAN_3:func.8">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>
 
<b>by </b><i><a class="txt" href="lopban_3.html#E4:73"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="series_1.html#T24" title="SERIES_1:th.24">SERIES_1:24</a></i>;<br/>
<a NAME="E7:73"/><b>then </b>
<font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K8" title="LOPBAN_3:func.8">GeoSeq</a>  is <a href="lopban_3.html#V2" title="LOPBAN_3:attr.2">norm_summable</a>
 
<b>by </b><i><a class="ref" href="lopban_3.html#D4" target="_self" title="LOPBAN_3:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E8:73"/>
( <font color="Maroon" title="c2">z</font> <a href="lopban_3.html#K8" title="LOPBAN_3:func.8">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#K8" title="LOPBAN_3:func.8">GeoSeq</a>  is <a href="lopban_3.html#V2" title="LOPBAN_3:attr.2">norm_summable</a> )
 <b>by </b><i><a class="ref" href="lopban_3.html#T31" target="_self" title="LOPBAN_3:th.31">Th31</a></i>;<br/>


</div>
