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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="clopban2.html#NM3" title="CLOPBAN2:NM.3">Complex_Banach_Algebra</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b1">z</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</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="clopban3.html#K2" title="CLOPBAN3:func.2">GeoSeq</a>  is  <a href="clopban3.html#V1" title="CLOPBAN3:attr.1">summable</a>  &amp; <font color="Olive" title="b1">z</font> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">GeoSeq</a>  is  <a href="clopban3.html#V2" title="CLOPBAN3: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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</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="clopban3.html#K2" title="CLOPBAN3:func.2">GeoSeq</a>  is  <a href="clopban3.html#V1" title="CLOPBAN3:attr.1">summable</a>  &amp; <font color="Maroon" title="c2">z</font> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">GeoSeq</a>  is  <a href="clopban3.html#V2" title="CLOPBAN3:attr.2">norm_summable</a>  ) )</span><br/>



<a NAME="E1:49"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#T105" onmouseover="rs('clvect_1/T105')" onmouseout="rh()">CLVECT_1:105</a></span>;<br/>
<a NAME="E2:49"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">GeoSeq</a></span>)</span></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> &amp; <span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">GeoSeq</a></span>)</span></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</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_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="49_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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span>(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">GeoSeq</a></span>)</span></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1 &amp; <span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">GeoSeq</a></span>)</span></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</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_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1 );<br/>
<a NAME="E1:49_1"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
<span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">GeoSeq</a></span>)</span></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</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="clopban3.html#K2" title="CLOPBAN3:func.2">GeoSeq</a></span>)</span> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_0.html#D4" onmouseover="rs('normsp_0/D4')" onmouseout="rh()">NORMSP_0:def 4</a></span>;<br/>
<a NAME="E2:49_1"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="49_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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<a NAME="E1:49_1_1"/><span class="lab"><font color="Green" title="E35">A5</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#T105" onmouseover="rs('clvect_1/T105')" onmouseout="rh()">CLVECT_1:105</a></span>;<br/>
<a NAME="E2:49_1_1"/>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</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="clopban3.html#K2" title="CLOPBAN3:func.2">GeoSeq</a></span>)</span> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</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_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</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="clopban3.html#K2" title="CLOPBAN3:func.2">GeoSeq</a></span>)</span> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban2.html#D9" onmouseover="rs('clopban2/D9')" onmouseout="rh()">CLOPBAN2:def 9</a></span>;<br/>
<a NAME="E3:49_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A6</font></span>: 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</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="clopban3.html#K2" title="CLOPBAN3:func.2">GeoSeq</a></span>)</span> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</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_0.html#K2" title="NORMSP_0:func.2">.||</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_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">GeoSeq</a></span>)</span></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_0.html#D4" onmouseover="rs('normsp_0/D4')" onmouseout="rh()">NORMSP_0:def 4</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:49_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E5:49_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">GeoSeq</a></span>)</span></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</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_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_1_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/>
<a NAME="E6:49_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A7</font></span>: 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</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="clopban3.html#K2" title="CLOPBAN3:func.2">GeoSeq</a></span>)</span> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</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_0.html#K2" title="NORMSP_0:func.2">.||</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_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_1_1"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">GeoSeq</a></span>)</span></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</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="clopban3.html#K2" title="CLOPBAN3:func.2">GeoSeq</a></span>)</span> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_0.html#D4" onmouseover="rs('normsp_0/D4')" onmouseout="rh()">NORMSP_0:def 4</a></span>
<br/>.= 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</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="clopban3.html#K2" title="CLOPBAN3:func.2">GeoSeq</a></span>)</span> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</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_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#D7" target="_self" onmouseover="rs('clopban3/D7')" onmouseout="rh()">Def4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:49_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:49_1_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="clvect_1.html#T105" onmouseover="rs('clvect_1/T105')" onmouseout="rh()">CLVECT_1:105</a>, <a class="ref" href="prepower.html#T3" onmouseover="rs('prepower/T3')" onmouseout="rh()">PREPOWER:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</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="clopban3.html#K2" title="CLOPBAN3:func.2">GeoSeq</a></span>)</span> <a href="normsp_0.html#K1" title="NORMSP_0:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> = 
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</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_0.html#K2" title="NORMSP_0:func.2">.||</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#D7" target="_self" onmouseover="rs('clopban3/D7')" onmouseout="rh()">Def4</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban2.html#D10" onmouseover="rs('clopban2/D10')" onmouseout="rh()">CLOPBAN2:def 10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#T3" onmouseover="rs('prepower/T3')" onmouseout="rh()">PREPOWER:3</a></span>
;<br/>
<a NAME="E4:49_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A8</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="clvect_1.html#T105" onmouseover="rs('clvect_1/T105')" onmouseout="rh()">CLVECT_1:105</a></span>;<br/>
<a NAME="E5:49_1"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E4:49_1"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="txt" href="#E2:49_1"><span class="lab"><font color="Green" title="E34">A4</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:49_1"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">GeoSeq</a></span>)</span></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> &amp; <span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">GeoSeq</a></span>)</span></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</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_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E3:49"/>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">z</font> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">GeoSeq</a>  is  <a href="clopban3.html#V1" title="CLOPBAN3:attr.1">summable</a>  &amp; <font color="Maroon" title="c2">z</font> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">GeoSeq</a>  is  <a href="clopban3.html#V2" title="CLOPBAN3:attr.2">norm_summable</a>  )</span><br/>

<a NAME="E4:49"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</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="#E1:49"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>;<br/>
<a NAME="E5:49"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Maroon" title="c2">z</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</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#T24" onmouseover="rs('series_1/T24')" onmouseout="rh()">SERIES_1:24</a></span>;<br/>
<a NAME="E6:49"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">GeoSeq</a></span>)</span></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</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="#E2:49"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="series_1.html#T20" onmouseover="rs('series_1/T20')" onmouseout="rh()">SERIES_1:20</a></span>;<br/>
<a NAME="E7:49"/><span class="kw">then </span>
<font color="Maroon" title="c2">z</font> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">GeoSeq</a>  is  <a href="clopban3.html#V2" title="CLOPBAN3:attr.2">norm_summable</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E8:49"/>
( <font color="Maroon" title="c2">z</font> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">GeoSeq</a>  is  <a href="clopban3.html#V1" title="CLOPBAN3:attr.1">summable</a>  &amp; <font color="Maroon" title="c2">z</font> <a href="clopban3.html#K2" title="CLOPBAN3:func.2">GeoSeq</a>  is  <a href="clopban3.html#V2" title="CLOPBAN3:attr.2">norm_summable</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
