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

<b>let </b><font color="Maroon">seq1</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">seq2</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:74"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">seq1</font> is <a href="series_1.html#V2">absolutely_summable</a>
 <b>and </b><br/><a NAME="E2:74"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">seq2</font> is <a href="series_1.html#V1">summable</a>
 ;<br/>

<b>set </b><font color="Maroon">S</font> = <font color="Maroon">seq1</font> <a href="catalan2.html#K6" target="_self">(##)</a> <font color="Maroon">seq2</font>;<br/>
<b>set </b><font color="Maroon">P1</font> =  <a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq1</font>;<br/>
<b>set </b><font color="Maroon">P2</font> =  <a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq2</font>;<br/>
<b>set </b><font color="Maroon">PA</font> =  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">P</font> =  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="catalan2.html#K6" target="_self">(##)</a> <font color="Maroon">seq2</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">S1</font> =  <a href="series_1.html#K2">Sum</a> <font color="Maroon">seq1</font>;<br/>
<b>set </b><font color="Maroon">S2</font> =  <a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font>;<br/>
<a NAME="E3:74"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">e</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e</font> holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">seq1</font> <a href="catalan2.html#K6" target="_self">(##)</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq1</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:74_1"/><i><font color="Green">E57</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 ;<br/>

<b>consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:74_1"/><i><font color="Green">E58</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="seqm_3.html#K1">^\</a> <font color="Olive">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> ) )
 <b>by </b><i><a class="ref" href="catalan2.html#T4" target="_self">Th4</a>, <a class="txt" href="catalan2.html#E4:74_1"><i><font color="Green">E59</font></i></a></i>;<br/>
<div><i><font color="Green">E59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:74_1_1"/>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a>, <a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><b>hence </b><a NAME="E2:74_1_1"/>
<span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon">e1</font> = <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">e2</font> = <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">3 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">e3</font> = <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span>;<br/>
<a NAME="E5:74_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E6:74_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="binop_2.html#K23">+</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E7:74_1"/><b>then </b><i><font color="Green">E60</font></i>: 
3 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="binop_2.html#K24">*</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E8:74_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/>
<a NAME="E9:74_1"/>
<font color="Maroon">seq1</font> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="txt" href="catalan2.html#E12:74_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="series_1.html#T40">SERIES_1:40</a></i>;<br/>
<a NAME="E10:74_1"/><b>then </b>
(  <a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq1</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <font color="Maroon">seq1</font> &amp; <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i>, <a class="ref" href="catalan2.html#D1" target="_self">Def1</a>, <a class="ref" href="series_1.html#D2">SERIES_1:def 2</a>, <a class="ref" href="series_1.html#D3">SERIES_1:def 3</a>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/>
<b>then consider </b><font color="Maroon">n0</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E12:74_1"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<a NAME="E13:74_1"/><i><font color="Green">E93</font></i>: 
 <a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="txt" href="catalan2.html#E12:74_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="series_1.html#D5">SERIES_1:def 5</a></i>;<br/>
<a NAME="E14:74_1"/>
3 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="binop_2.html#K24">*</a> 3
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E15:74_1"/><b>then </b>
<font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">3 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/>
<b>then consider </b><font color="Maroon">n1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E17:74_1"/><i><font color="Green">E95</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">3 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="catalan2.html#E9"><i><font color="Green">Lemma71</font></i></a>, <a class="ref" href="series_1.html#T25">SERIES_1:25</a></i>;<br/>
<a NAME="E18:74_1"/>
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="catalan2.html#E9"><i><font color="Green">Lemma71</font></i></a>, <a class="ref" href="catalan2.html#T5" target="_self">Th5</a>, <a class="ref" href="series_1.html#T21">SERIES_1:21</a></i>;<br/>
<a NAME="E19:74_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="binop_2.html#K23">+</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E20:74_1"/><b>then </b>
3 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="binop_2.html#K24">*</a> 3
 
<b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E21:74_1"/><b>then </b><i><font color="Green">E96</font></i>: 
( <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq2</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font> )
 
<b>by </b><i><a class="ref" href="catalan2.html#T4" target="_self">Th4</a>, <a class="ref" href="series_1.html#D2">SERIES_1:def 2</a>, <a class="ref" href="series_1.html#D3">SERIES_1:def 3</a>, , <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/>
<b>then consider </b><font color="Maroon">n2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E23:74_1"/><i><font color="Green">E97</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<a NAME="E24:74_1"/><i><font color="Green">E98</font></i>: 
(  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1 or  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font> )
 
<b>by </b><i><a class="ref" href="square_1.html#T49">SQUARE_1:49</a></i>;<br/>
<a NAME="E25:74_1"/>
(  <a href="square_1.html#K2">max</a> 1,<font color="Maroon">n0</font> <a href="hidden.html#R1">=</a> 1 or  <a href="square_1.html#K2">max</a> 1,<font color="Maroon">n0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n0</font> )
 
<b>by </b><i><a class="ref" href="square_1.html#T49">SQUARE_1:49</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">M</font> =  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><a href="square_1.html#K2">max</a> 1,<font color="Maroon">n0</font></span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K2">max</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">n2</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="catalan2.html#E14"><i><font color="Green">Lemma75</font></i></a>, <a class="ref" href="square_1.html#T49">SQUARE_1:49</a></i>;<br/>
<b>take </b><font color="Maroon">2M</font> = <font color="Maroon">M</font> <a href="binop_2.html#K24">*</a> 2;<br/>

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E27:74_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">2M</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E28:74_1"/><i><font color="Green">E129</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K2">max</a> 1,<font color="Maroon">n0</font> &amp; <font color="Maroon">n0</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K2">max</a> 1,<font color="Maroon">n0</font> &amp;  <a href="square_1.html#K2">max</a> 1,<font color="Maroon">n0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> &amp; <font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">n2</font> &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">n2</font> &amp;  <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> &amp; <font color="Maroon">M</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">M</font> )
 
<b>by </b><i><a class="ref" href="square_1.html#T46">SQUARE_1:46</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E29:74_1"/><b>then </b><i><font color="Green">E131</font></i>: 
( <font color="Maroon">n0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> &amp; <font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">M</font> &amp; <font color="Maroon">M</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 
<b>by </b><i><a class="ref" href="catalan2.html#T9" target="_self">Th9</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">M1</font> = <font color="Maroon">M</font> <a href="binop_2.html#K10">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="nat_1.html#T60">NAT_1:60</a></i>;<br/>
<a NAME="E31:74_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E32:74_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="binop_2.html#K23">+</a> 0 &amp; <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 0 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E33:74_1"/><b>then </b><i><font color="Green">E132</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p5">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 3 &amp; <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p5">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p5">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="catalan2.html#T6" target="_self">Th6</a>, <a class="ref" href="xcmplx_1.html#T93">XCMPLX_1:93</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E34:74_1"/>
<font color="Maroon">n0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="catalan2.html#T10" target="_self">Th10</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E35:74_1"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq1</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i><a class="ref" href="catalan2.html#T7" target="_self">Th7</a>, <a class="ref" href="complex1.html#T151">COMPLEX1:151</a>, <a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E36:74_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq1</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p5">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E37:74_1"/><b>then </b><i><font color="Green">E133</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq1</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 3
 
<b>by </b><i><a class="ref" href="catalan2.html#T11" target="_self">Th11</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<b>consider </b><font color="Maroon">Fr</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E39:74_1"/><i><font color="Green">E134</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="catalan2.html#K6" target="_self">(##)</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr</font></span>)</span>
 <b>and </b><br/><a NAME="E40:74_1"/><i><font color="Green">E135</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1
 <b>and </b><br/><a NAME="E41:74_1"/><i><font color="Green">E136</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1 holds <br/><font color="Maroon">Fr</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> <font color="Olive">i</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="catalan2.html#T4" target="_self">Th4</a>, <a class="txt" href="catalan2.html#E1:74_1"><i><font color="Green">E57</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">absFr</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E43:74_1"/><i><font color="Green">E139</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">absFr</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr</font>
 <b>and </b><br/><a NAME="E44:74_1"/><i><font color="Green">E140</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">absFr</font>
 <b>and </b><br/><a NAME="E45:74_1"/><i><font color="Green">E141</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">absFr</font> holds <br/><font color="Maroon">absFr</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">Fr</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Olive">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="catalan2.html#E3:74_1"><i><font color="Green">E58</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> implies  <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> );<br/>
<a NAME="E46:74_1"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:74_1_2"/>
(  <a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 0</span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 0</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 
<b>by </b><i><a class="ref" href="catalan2.html#T4" target="_self">Th4</a>, <a class="ref" href="series_1.html#T18">SERIES_1:18</a>, <a class="ref" href="jordan3.html#T2">JORDAN3:2</a></i>;<br/>
<a NAME="E2:74_1_2"/><b>then </b><i><font color="Green">E144</font></i>: 
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 0</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
 
;<br/>
<a NAME="E3:74_1_2"/>
( 0 <a href="hidden.html#R2">in</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1 &amp;  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="relat_1.html#T110">RELAT_1:110</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>
<a NAME="E4:74_1_2"/><b>then </b>
(  <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">absFr</font> <a href="catalan2.html#K5" target="_self">.</a> 0</span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> 0</span>)</span></span>)</span> &amp;  <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">absFr</font> <a href="catalan2.html#K5" target="_self">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">Fr</font> <a href="catalan2.html#K5" target="_self">.</a> 0</span>)</span> &amp; <font color="Maroon">Fr</font> <a href="catalan2.html#K5" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 0</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="catalan2.html#T14" target="_self">Th14</a>, <a class="ref" href="catalan2.html#T15" target="_self">Th15</a>, <a class="ref" href="catalan2.html#T17" target="_self">Th17</a>, , , <a class="txt" href="catalan2.html#E1:74"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>then </b><i><font color="Green">E148</font></i>:  <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> 0</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">seq2</font> <a href="seqm_3.html#K1">^\</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 0</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">seq2</font> <a href="seqm_3.html#K1">^\</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 0</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="catalan2.html#T18" target="_self">Th18</a>, <a class="ref" href="complex1.html#T146">COMPLEX1:146</a></i>
;<br/>
<a NAME="E6:74_1_2"/>
( <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> &amp; <font color="Maroon">M</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">M</font> )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E7:74_1_2"/><b>then </b>
<font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">2M</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E8:74_1_2"/><b>then </b>
<font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="catalan2.html#T9" target="_self">Th9</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E9:74_1_2"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i><a class="txt" href="catalan2.html#E13"><i><font color="Green">Lemma74</font></i></a>, <a class="ref" href="catalan2.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E10:74_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> 0</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<b>hence </b><a NAME="E11:74_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i>, <a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E47:74_1"/><i><font color="Green">E149</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:74_1_3"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>set </b><font color="Maroon">k1</font> = <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1;<br/>
<b>assume </b><a NAME="E2:74_1_3"/><i><font color="Green">E151</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font>
 ;<br/>

<a NAME="E3:74_1_3"/><b>then </b><i><font color="Green">E152</font></i>: 
<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:74_1_3"/><b>then </b><i><font color="Green">E153</font></i>: 
( <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="ref" href="catalan2.html#T10" target="_self">Th10</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:74_1_3"/><b>then </b>
<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E6:74_1_3"/><b>then </b>
<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E7:74_1_3"/><b>then </b><i><font color="Green">E154</font></i>: 
( <font color="Maroon">Fr</font> <a href="catalan2.html#K5" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">Fr</font> <a href="catalan2.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">absFr</font> <a href="catalan2.html#K5" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> &amp;  <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">absFr</font> <a href="catalan2.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="catalan2.html#T14" target="_self">Th14</a>, <a class="ref" href="catalan2.html#T15" target="_self">Th15</a>, <a class="ref" href="catalan2.html#T17" target="_self">Th17</a>, <a class="txt" href="catalan2.html#E1:74"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E8:74_1_3"/><i><font color="Green">E155</font></i>: 
 <a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="catalan2.html#T4" target="_self">Th4</a>, <a class="ref" href="series_1.html#T18">SERIES_1:18</a></i>;<br/>
<a NAME="E9:74_1_3"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p0">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p0">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p0">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="seq_1.html#T16">SEQ_1:16</a>, , , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E10:74_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p0">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E11:74_1_3"/><b>then </b><i><font color="Green">E156</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p0">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>;<br/>
<a NAME="E12:74_1_3"/>
( <font color="Maroon">m</font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">M</font> &amp; <font color="Maroon">m</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">M</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">2M</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">M</font> )
 
<b>by </b><i><a class="ref" href="catalan2.html#T9" target="_self">Th9</a>, , <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a>, <a class="ref" href="xreal_1.html#T12">XREAL_1:12</a></i>;<br/>
<a NAME="E13:74_1_3"/><b>then </b>
<font color="Maroon">m</font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E14:74_1_3"/><b>then </b>
( <font color="Maroon">m</font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n2</font> &amp; <font color="Maroon">m</font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> )
 
<b>by </b><i>, <a class="ref" href="catalan2.html#T10" target="_self">Th10</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E15:74_1_3"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="catalan2.html#E13"><i><font color="Green">Lemma74</font></i></a></i>;<br/>
<a NAME="E16:74_1_3"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i><a class="ref" href="complex1.html#T146">COMPLEX1:146</a>, <a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E17:74_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E18:74_1_3"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">seq2</font> <a href="seqm_3.html#K1">^\</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="catalan2.html#T19" target="_self">Th19</a>, <a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>;<br/>
<a NAME="E19:74_1_3"/><b>then </b>
 <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p0">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="catalan2.html#E13:74_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E20:74_1_3"/>
 <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="catalan2.html#E17:74_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E48:74_1"/><i><font color="Green">E157</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, <a class="ref" href="catalan2.html#T18" target="_self">Th18</a>);<br/></i>
<a NAME="E49:74_1"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">M1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="catalan2.html#T5" target="_self">Th5</a>, <a class="txt" href="catalan2.html#E9"><i><font color="Green">Lemma71</font></i></a>, <a class="ref" href="rsspace2.html#T4">RSSPACE2:4</a></i>;<br/>
<a NAME="E50:74_1"/><b>then </b><i><font color="Green">E158</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">M1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="catalan2.html#E11"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E51:74_1"/>
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="catalan2.html#E9"><i><font color="Green">Lemma71</font></i></a>, <a class="ref" href="catalan2.html#T5" target="_self">Th5</a>, <a class="ref" href="series_1.html#T21">SERIES_1:21</a></i>;<br/>
<a NAME="E52:74_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="binop_2.html#K23">+</a> 0 &amp; <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 0 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E53:74_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 3 &amp; <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> &amp; <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M1</font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="txt" href="catalan2.html#E11"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="xcmplx_1.html#T93">XCMPLX_1:93</a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E54:74_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">M1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 3 &amp;  <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">M</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p5">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">M1</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E55:74_1"/><b>then </b><i><font color="Green">E159</font></i>: 
 <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">M</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 3
 
<b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<b>consider </b><font color="Maroon">Fr1</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E57:74_1"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">absFr</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">M</font></span>)</span> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">Fr1</font>
 <b>by </b><i/>;<br/>
<a NAME="E58:74_1"/>
<font color="Maroon">M</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="catalan2.html#T10" target="_self">Th10</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E59:74_1"/><b>then </b>
<font color="Maroon">M</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">absFr</font>
 
<b>by </b><i>, <a class="ref" href="catalan2.html#T15" target="_self">Th15</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E60:74_1"/><b>then </b><i><font color="Green">E161</font></i>: 
( <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <span class="p2">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font></span>)</span> &amp;  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> )
 
<b>by </b><i>, <a class="ref" href="catalan2.html#T15" target="_self">Th15</a>, , <a class="ref" href="afinsq_1.html#D4">AFINSQ_1:def 4</a>, <a class="ref" href="afinsq_1.html#T14">AFINSQ_1:14</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1 implies  <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Fr1</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">M1</font></span>)</span></span>)</span> );<br/>
<a NAME="E61:74_1"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:74_1_4"/><i><font color="Green">E163</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> 0</span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/>

<a NAME="E2:74_1_4"/>
( <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">M</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">M</font> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">M</font> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E13:74_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="catalan2.html#E17:74_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E3:74_1_4"/><b>then </b>
( 1 <a href="tarski.html#R1">c=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr1</font> )
 
<b>by </b><i><a class="ref" href="card_1.html#T56">CARD_1:56</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E4:74_1_4"/><b>then </b>
(  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fr1</font> <a href="card_fin.html#K4">|</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; 0 <a href="hidden.html#R2">in</a> 1 )
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E5:74_1_4"/><b>then </b><i><font color="Green">E164</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">Fr1</font> <a href="card_fin.html#K4">|</a> 1</span>)</span> <a href="catalan2.html#K5" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">Fr1</font> <a href="catalan2.html#K5" target="_self">.</a> 0 &amp;  <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Fr1</font> <a href="card_fin.html#K4">|</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Fr1</font> <a href="card_fin.html#K4">|</a> 1</span>)</span> <a href="catalan2.html#K5" target="_self">.</a> 0 )
 
<b>by </b><i><a class="ref" href="catalan2.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E6:74_1_4"/>
( <font color="Maroon">M</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">absFr</font> )
 
<b>by </b><i>, <a class="ref" href="catalan2.html#T15" target="_self">Th15</a>, <a class="txt" href="catalan2.html#E17:74_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E7:74_1_4"/><b>then </b>
( <font color="Maroon">absFr</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fr1</font> <a href="catalan2.html#K5" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">M</font></span>)</span> &amp; <font color="Maroon">M</font> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="txt" href="catalan2.html#E13:74_1"><i><font color="Green">E93</font></i></a>, , <a class="ref" href="afinsq_1.html#T22">AFINSQ_1:22</a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E8:74_1_4"/><b>then </b>
( <font color="Maroon">Fr1</font> <a href="catalan2.html#K5" target="_self">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">Fr</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">M</font></span>)</span> &amp; <font color="Maroon">Fr</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">M</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">M</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="catalan2.html#T14" target="_self">Th14</a>, <a class="ref" href="catalan2.html#T15" target="_self">Th15</a>, <a class="ref" href="catalan2.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E9:74_1_4"/><b>then </b><i><font color="Green">E165</font></i>: 
<font color="Maroon">Fr1</font> <a href="catalan2.html#K5" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">seq2</font> <a href="seqm_3.html#K1">^\</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">M</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>;<br/>
<a NAME="E10:74_1_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">M1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>;<br/>
<a NAME="E11:74_1_4"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">M</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">M1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">M</font></span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">M</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">M</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="complex1.html#T132">COMPLEX1:132</a>, <a class="ref" href="seq_1.html#T16">SEQ_1:16</a></i>;<br/>
<b>hence </b><a NAME="E12:74_1_4"/>
 <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Fr1</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> 0</span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">M1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="catalan2.html#E21:74_1"><i><font color="Green">E96</font></i></a>, <a class="txt" href="catalan2.html#E23:74_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E62:74_1"/><i><font color="Green">E166</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:74_1_5"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>set </b><font color="Maroon">k1</font> = <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1;<br/>
<b>set </b><font color="Maroon">Mk1</font> = <font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>;<br/>
<b>assume </b><a NAME="E2:74_1_5"/><i><font color="Green">E168</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/>

<a NAME="E3:74_1_5"/><b>then </b>
( <font color="Maroon">M</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> &amp; <font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">absFr</font> )
 
<b>by </b><i>, <a class="ref" href="catalan2.html#T15" target="_self">Th15</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E4:74_1_5"/><b>then </b>
( <font color="Maroon">absFr</font> <a href="catalan2.html#K5" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Fr1</font> <a href="catalan2.html#K5" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">M</font></span>)</span> &amp; <font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i>, <a class="txt" href="catalan2.html#E13:74_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="afinsq_1.html#T22">AFINSQ_1:22</a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E5:74_1_5"/><b>then </b><i><font color="Green">E169</font></i>: 
( <font color="Maroon">Fr1</font> <a href="catalan2.html#K5" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">Fr</font> <a href="catalan2.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">Fr</font> <a href="catalan2.html#K5" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="catalan2.html#T15" target="_self">Th15</a>, <a class="ref" href="catalan2.html#T14" target="_self">Th14</a>, <a class="ref" href="catalan2.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E6:74_1_5"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">Fr</font> <a href="catalan2.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">seq2</font> <a href="seqm_3.html#K1">^\</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> )
 
<b>by </b><i>, <a class="ref" href="complex1.html#T132">COMPLEX1:132</a>, <a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>;<br/>
<a NAME="E7:74_1_5"/><b>then </b><i><font color="Green">E170</font></i>: 
<font color="Maroon">Fr1</font> <a href="catalan2.html#K5" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="catalan2.html#E24:74_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E8:74_1_5"/>
<font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="catalan2.html#E23:74_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E9:74_1_5"/><b>then </b>
<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font>
 
<b>by </b><i><a class="txt" href="catalan2.html#E13:74_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E10:74_1_5"/><b>then </b>
( <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr1</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font> )
 
<b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E11:74_1_5"/><b>then </b>
(  <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Fr1</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Fr1</font> <a href="catalan2.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">Fr1</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> &amp;  <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Fr1</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">M1</font></span>)</span></span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E21:74_1"><i><font color="Green">E96</font></i></a>, <a class="txt" href="catalan2.html#E1:74"><i><font color="Green">E51</font></i></a>, <a class="ref" href="seq_1.html#T16">SEQ_1:16</a>, <a class="txt" href="catalan2.html#E23:74_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E12:74_1_5"/><b>then </b>
 <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Fr1</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">M1</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="catalan2.html#E27:74_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E13:74_1_5"/><b>then </b>
 <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Fr1</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p5">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">M1</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E14:74_1_5"/><b>then </b>
 <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Fr1</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">M1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E15:74_1_5"/>
 <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Fr1</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="binop_2.html#K23">+</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">M1</font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E63:74_1"/><i><font color="Green">E171</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="catalan2.html#T19" target="_self">Th19</a>, <a class="txt" href="catalan2.html#E17:74_1"><i><font color="Green">E95</font></i></a>);<br/></i>
<a NAME="E64:74_1"/>
<font color="Maroon">M1</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E65:74_1"/><b>then </b>
<font color="Maroon">M1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n1</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E66:74_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">M1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">n1</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 
<b>by </b><i><a class="ref" href="catalan2.html#T10" target="_self">Th10</a>, <a class="ref" href="catalan2.html#T5" target="_self">Th5</a>, <a class="txt" href="catalan2.html#E7:74_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E67:74_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">M1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font></span>)</span> &amp; <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T12">XREAL_1:12</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E68:74_1"/><b>then </b><i><font color="Green">E172</font></i>: 
( <font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">3 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="catalan2.html#T5" target="_self">Th5</a>, , <a class="txt" href="catalan2.html#E10"><i><font color="Green">Lemma72</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a>, <a class="txt" href="catalan2.html#E7:74_1"><i><font color="Green">E60</font></i></a></i>;<br/>
<a NAME="E69:74_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E70:74_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E71:74_1"/><b>then </b>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">3 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="catalan2.html#E23:74_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E72:74_1"/><b>then </b><i><font color="Green">E173</font></i>: 
<font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">3 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="catalan2.html#E23:74_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E73:74_1"/>
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">M</font></span>)</span> <a href="binop_2.html#K9">+</a> 1
 
<b>by </b><i><a class="txt" href="catalan2.html#E13:74_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E74:74_1"/><b>then </b>
( <font color="Maroon">Fr1</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">M</font></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Fr1</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">M</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">m</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">M</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T98">RELAT_1:98</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a>, <a class="ref" href="catalan2.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E75:74_1"/><b>then </b>
 <a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><a href="seq_1.html#K22">abs</a> <font color="Maroon">seq1</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">M1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="catalan2.html#E21:74_1"><i><font color="Green">E96</font></i></a></i>;<br/>
<a NAME="E76:74_1"/><b>then </b>
 <a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">3 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="catalan2.html#E24:74_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E77:74_1"/><b>then </b><i><font color="Green">E174</font></i>: 
 <a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 3
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T93">XCMPLX_1:93</a></i>;<br/>
<a NAME="E78:74_1"/>
 <a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">absFr</font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">M</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="stirl2_1.html#T47">STIRL2_1:47</a></i>;<br/>
<a NAME="E79:74_1"/><b>then </b>
(  <a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">absFr</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr1</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 3</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 3</span>)</span> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E27:74_1"><i><font color="Green">E99</font></i></a>, , <a class="ref" href="binop_2.html#D9">BINOP_2:def 9</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E80:74_1"/><b>then </b><i><font color="Green">E175</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 3</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 3</span>)</span>
 
<b>by </b><i><a class="ref" href="catalan2.html#T16" target="_self">Th16</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E81:74_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">seq1</font> <a href="catalan2.html#K6" target="_self">(##)</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq1</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr</font></span>)</span>
 
<b>by </b><i><a class="ref" href="catalan2.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E82:74_1"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">seq1</font> <a href="catalan2.html#K6" target="_self">(##)</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq1</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 3</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 3</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 3</span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="catalan2.html#T12" target="_self">Th12</a>, <a class="ref" href="catalan2.html#T20" target="_self">Th20</a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a>, <a class="ref" href="complex1.html#T143">COMPLEX1:143</a></i>;<br/>
<b>hence </b><a NAME="E83:74_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p4">(<span class="default"><font color="Maroon">seq1</font> <a href="catalan2.html#K6" target="_self">(##)</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq1</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:74"/><b>then </b><i><font color="Green">E177</font></i>: 
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="catalan2.html#K6" target="_self">(##)</a> <font color="Maroon">seq2</font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E5:74"/>
<font color="Maroon">seq1</font> <a href="catalan2.html#K6" target="_self">(##)</a> <font color="Maroon">seq2</font> is <a href="series_1.html#V1">summable</a>
 <b>by </b><i><a class="ref" href="series_1.html#D2">SERIES_1:def 2</a></i>;<br/>

<a NAME="E6:74"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="catalan2.html#K6" target="_self">(##)</a> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq1</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E7:74"/>
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="catalan2.html#K6" target="_self">(##)</a> <font color="Maroon">seq2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq1</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span>
 <b>by </b><i><a class="ref" href="series_1.html#D3">SERIES_1:def 3</a></i>;<br/>


</div>
