<?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>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/>





<b>assume </b><a NAME="E1:70"/><i><font color="Green">E51</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>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>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#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></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">n</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>;<br/>
<a NAME="E2:70"/><i><font color="Green">E53</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">n</font> <a href="binop_2.html#K23">+</a> 1 holds <br/> ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:70_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/>

<b>take </b>
<span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">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">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
;<br/>

<b>thus </b><a NAME="E2:70_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">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">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>]
 ;<br/>


</div><b>end;</b></div>
<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="E4:70"/><i><font color="Green">E54</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 <b>and </b><br/><a NAME="E5:70"/><i><font color="Green">E57</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">n</font> <a href="binop_2.html#K23">+</a> 1 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Maroon">Fr</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="stirl2_1.html#S6">STIRL2_1:sch 6</a>(<a class="txt" href="catalan2.html#E8:70"><i><font color="Green">E59</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">Fr</font>
;<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="E7:70"/><i><font color="Green">E58</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">n</font> <a href="hidden.html#R1">=</a>  <a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr1</font>
 <b>and </b><br/><a NAME="E8:70"/><i><font color="Green">E59</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 <b>and </b><br/><a NAME="E9:70"/><i><font color="Green">E60</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">n</font> <a href="binop_2.html#K23">+</a> 1 holds <br/><font color="Maroon">Fr1</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"><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">n</font> <a href="binarith.html#K3">-'</a> <font color="Olive">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="catalan2.html#T3" target="_self">Th3</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>( <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">n</font> <a href="binop_2.html#K23">+</a> 1 implies <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">a<sub>1</sub></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="catalan2.html#K4" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">Fr</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></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="series_1.html#K2">Sum</a> <font color="Maroon">seq2</font></span>)</span> <a href="binop_2.html#K11">*</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">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> );<br/>
<a NAME="E10:70"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2"><b>proof </b></a><div class="add">

<a NAME="E1:70_2"/>
( 0 <a href="hidden.html#R2">in</a> <font color="Maroon">n</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">Fr</font> <a href="card_fin.html#K4">|</a> 0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <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> 0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a>, <a class="ref" href="relat_1.html#T110">RELAT_1:110</a></i>;<br/>
<a NAME="E2:70_2"/><b>then </b>
(  <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Fr</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">Fr</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">Fr</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">Fr</font> <a href="card_fin.html#K4">|</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 0 &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">n</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">Fr1</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"><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">n</font> <a href="binarith.html#K3">-'</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">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="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Fr1</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">Fr1</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">Fr1</font> <a href="card_fin.html#K4">|</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="catalan2.html#E9:70"><i><font color="Green">E60</font></i></a>, <a class="txt" href="catalan2.html#E10:70"><i><font color="Green">E61</font></i></a>, <a class="ref" href="catalan2.html#T4" target="_self">Th4</a>, , <a class="txt" href="catalan2.html#E1:70"><i><font color="Green">E51</font></i></a>, </i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">Fr</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default">0 <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="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">0 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<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"><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">n</font> <a href="binarith.html#K3">-'</a> 0</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></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> <font color="Maroon">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 0</span>)</span></span>)</span></span>)</span>

<br/>.= 
<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> <font color="Maroon">seq2</font></span>)</span>
<b>by </b><i><a class="txt" href="catalan2.html#E7:70"><i><font color="Green">E58</font></i></a>, <a class="ref" href="series_1.html#T18">SERIES_1:18</a></i>
;<br/>
<b>hence </b><a NAME="E4:70_2"/>
<font color="Maroon">S<sub>2</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="E11:70"/><i><font color="Green">E62</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="70_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="E1:70_3"/><i><font color="Green">E93</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>assume </b><a NAME="E2:70_3"/>
<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">n</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/>

<a NAME="E3:70_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">n</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <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 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:70_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">n</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</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="E5:70_3"/><b>then </b>
(  <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Fr</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">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="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">Fr</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; <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> <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"><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">n</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> &amp; <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">n</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="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; <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> <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">Fr</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="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#K11">*</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">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E9:70"><i><font color="Green">E60</font></i></a>, <a class="txt" href="catalan2.html#E10:70"><i><font color="Green">E61</font></i></a>, <a class="ref" href="catalan2.html#T4" target="_self">Th4</a>, , <a class="ref" href="catalan2.html#T5" target="_self">Th5</a>, <a class="txt" href="catalan2.html#E1:70"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">Fr</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<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></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"><span class="p4">(<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></span>)</span> = 
<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"><span class="p3">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p4">(<span class="default"><font color="Maroon">seq2</font> <a href="seqm_3.html#K1">^\</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</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></span>)</span> <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> <font color="Maroon">seq2</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</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#K9">+</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">k</font></span>)</span></span>)</span>

<br/>.= 
<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> <font color="Maroon">seq2</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</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">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="catalan2.html#E7:70"><i><font color="Green">E58</font></i></a>, <a class="ref" href="series_1.html#T18">SERIES_1:18</a></i>
<br/>.= 
<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#K11">*</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">k</font></span>)</span> <a href="binop_2.html#K9">+</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>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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> <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> <font color="Maroon">seq2</font></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="E7:70_3"/>
<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"><span class="p4">(<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></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">Fr</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><span class="p4">(<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></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#K11">*</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">seq1</font></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>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E12:70"/><i><font color="Green">E95</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>(, );<br/></i>
<a NAME="E13:70"/>
( <font color="Maroon">Fr</font> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Fr</font> &amp; <font color="Maroon">Fr1</font> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Fr1</font> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E9:70"><i><font color="Green">E60</font></i></a>, <a class="ref" href="catalan2.html#T4" target="_self">Th4</a>, <a class="ref" href="relat_1.html#T98">RELAT_1:98</a></i>;<br/>
<a NAME="E14:70"/><b>then </b>
<span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr1</font></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">Fr</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">seq2</font></span>)</span> <a href="binop_2.html#K11">*</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">seq1</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="catalan2.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E15:70"/>
( <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">n</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">n</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> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 &amp; ( 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">n</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">n</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="txt" href="catalan2.html#E9:70"><i><font color="Green">E60</font></i></a>, <a class="txt" href="catalan2.html#E10:70"><i><font color="Green">E61</font></i></a>, <a class="txt" href="catalan2.html#E11:70"><i><font color="Green">E62</font></i></a></i>;<br/>


</div>
