<?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>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">P</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="nat_1.html#NM1">Nat</a>] <b>means </b> ex <font color="Olive">Fr</font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <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">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalan2.html#K4" target="_self">Sum</a> <font color="Olive">Fr</font> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Olive">Fr</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</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">a<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 holds <br/><font color="Olive">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"><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">a<sub>1</sub></font> <a href="binarith.html#K3">-'</a> <font color="Olive">i</font></span>)</span></span>)</span> ) );<br/>
<a NAME="E1:69"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">Fr</font> = 1 <a href="funcop_1.html#K2">--&gt;</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> 0</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> 0</span>)</span></span>)</span>;<br/>
<a NAME="E1:69_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">1 <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default"><font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> 0</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Fr</font> = 1 <a href="funcop_1.html#K2">--&gt;</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> 0</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> 0</span>)</span></span>)</span> as   <a href="afinsq_1.html#NM1">XFinSequence</a> <b>by </b><i><a class="ref" href="afinsq_1.html#T7">AFINSQ_1:7</a></i>;<br/>
<a NAME="E3:69_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fr</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default"><font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> 0</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E4:69_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">Fr</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Fr</font> = <font color="Maroon">Fr</font> as   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D8">ORDINAL1:def 8</a></i>;<br/>
<b>take </b>
<font color="Maroon">Fr</font>
;<br/>

<a NAME="E6:69_1"/>
(  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr</font> <a href="hidden.html#R1">=</a> 1 &amp; 0 <a href="hidden.html#R2">in</a> 1 &amp; <font color="Maroon">Fr</font> <a href="card_fin.html#K4">|</a> 0 <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="relat_1.html#T110">RELAT_1:110</a></i>;<br/>
<a NAME="E7:69_1"/><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> 0</span>)</span></span>)</span> <a href="binop_2.html#K9">+</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="hidden.html#R1">=</a>  <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> &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"><font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> 0</span>)</span> &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> 0 &amp; <font color="Maroon">Fr</font> <a href="card_fin.html#K4">|</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">Fr</font> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E1:69"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a>, <a class="ref" href="relat_1.html#T98">RELAT_1:98</a></i>;<br/>
<a NAME="E8:69_1"/><b>then </b>
 <a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr</font> <a href="hidden.html#R1">=</a> 0 <a href="binop_2.html#K9">+</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> 0</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> 0</span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E9:69_1"/><b>then </b>
 <a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr</font> <a href="hidden.html#R1">=</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="seq_1.html#K2">.</a> 0
 
<b>by </b><i><a class="ref" href="catalan2.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E10:69_1"/>
( <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> 0 <a href="hidden.html#R1">=</a>  <a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr</font> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr</font> <a href="hidden.html#R1">=</a> 0 <a href="binop_2.html#K23">+</a> 1 )
 <b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>

<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="E11:69_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> 0 <a href="binop_2.html#K23">+</a> 1
 ;<br/>

<a NAME="E12:69_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 
<b>by </b><i>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E13:69_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E14:69_1"/><b>then </b>
( 0 <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr</font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E15:69_1"/><b>then </b>
( <span class="p1">(<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="p1">(<span class="default">0 <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</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"><font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> 0</span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="series_1.html#D1">SERIES_1:def 1</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>hence </b><a NAME="E16:69_1"/>
<font color="Maroon">Fr</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">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="Maroon">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">0 <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:69"/><i><font color="Green">E57</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">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2"><b>proof </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/>

<b>set </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1;<br/>
<b>assume </b><a NAME="E1:69_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<b>then 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="E3:69_2"/><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">Fr</font>
 <b>and </b><br/><a NAME="E4:69_2"/><i><font color="Green">E59</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="E5:69_2"/><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">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"><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>
 ;<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="E7:69_2"/><i><font color="Green">E61</font></i>: 
 <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">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1
 <b>and </b><br/><a NAME="E8:69_2"/><i><font color="Green">E62</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> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <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"><font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Olive">i</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E9:69_2"/><i><font color="Green">E93</font></i>: 
 <a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr1</font> <a href="hidden.html#R1">=</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="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</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"><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"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Olive">i</font></span>)</span></span>)</span>;<br/>
<a NAME="E10:69_2"/><i><font color="Green">E95</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> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <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>2</sub></font>[<font color="Olive">i</font>,<font color="Olive">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2_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="E1:69_2_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <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"><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"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span></span>)</span>
;<br/>

<b>thus </b><a NAME="E2:69_2_1"/>
<font color="Maroon">S<sub>2</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"><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"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span></span>)</span>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">Fr2</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="E12:69_2"/><i><font color="Green">E96</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1
 <b>and </b><br/><a NAME="E13:69_2"/><i><font color="Green">E97</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> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>,<font color="Maroon">Fr2</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>();<br/></i>
<b>take </b>
<font color="Maroon">Fr2</font>
;<br/>

<a NAME="E14:69_2"/><i><font color="Green">E98</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> <span class="p1">(<span class="default"><font color="Maroon">Fr2</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">Fr2</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="catalan2.html#K5" target="_self">.</a> <font color="Olive">i</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"><font color="Maroon">Fr</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p1">(<span class="default"><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">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="catalan2.html#K5" target="_self">.</a> <font color="Olive">i</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2_2"><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="E1:69_2_2"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fr2</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:69_2_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr2</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="catalan2.html#T6" target="_self">Th6</a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E3:69_2_2"/><b>then </b><i><font color="Green">E129</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 &amp; <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 &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E8:69_2"><i><font color="Green">E62</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E4:69_2_2"/><b>then </b><i><font color="Green">E131</font></i>: 
( <font color="Maroon">Fr1</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">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="Maroon">i</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span></span>)</span> &amp; <font color="Maroon">Fr</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">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="Maroon">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="Maroon">i</font></span>)</span></span>)</span> &amp; <font color="Maroon">Fr2</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">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="Maroon">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"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E7:69_2"><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></i>;<br/>
<a NAME="E5:69_2_2"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="ref" href="catalan2.html#T7" target="_self">Th7</a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E6:69_2_2"/><b>then </b>
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E7:69_2_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">i</font> &amp; <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">i</font> )
 
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E8:69_2_2"/><b>then </b>
<span class="p1">(<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 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font>
 
;<br/>
<a NAME="E9:69_2_2"/><b>then </b>
<span class="p1">(<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="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></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> <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="Maroon">i</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></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="E10:69_2_2"/><b>then </b>
( <font color="Maroon">Fr2</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">i</font> <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> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">Fr1</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">Fr1</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</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">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">Fr2</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Fr2</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">i</font> )
 
<b>by </b><i><a class="ref" href="catalan2.html#T6" target="_self">Th6</a>, <a class="ref" href="catalan2.html#T7" target="_self">Th7</a>, <a class="txt" href="catalan2.html#E9"><i><font color="Green">Lemma71</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E11:69_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">Fr2</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">i</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"><font color="Maroon">Fr</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><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">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="binop_2.html#D9">BINOP_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">A</font> =  <a href="binop_2.html#K33">addreal</a> ;<br/>
<a NAME="E15:69_2"/>
<font color="Maroon">n</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">n</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#T29">NAT_1:29</a></i>;<br/>
<a NAME="E16:69_2"/><b>then </b>
<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">n</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="card_1.html#T56">CARD_1:56</a></i>;<br/>
<a NAME="E17:69_2"/><b>then </b>
(  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fr2</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</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">Fr1</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr</font> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E8:69_2"><i><font color="Green">E62</font></i></a>, , <a class="ref" href="relat_1.html#T91">RELAT_1:91</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E18:69_2"/><b>then </b>
(  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">Fr2</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr</font> &amp;  <a href="afinsq_1.html#K1">len</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">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr</font> )
 
<b>by </b><i><a class="txt" href="catalan2.html#E5:69_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E19:69_2"/><b>then </b>
<a href="binop_2.html#K33">addreal</a>  <a href="stirl2_1.html#K4">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">Fr2</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K33">addreal</a>  <a href="stirl2_1.html#K4">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">Fr</font> <a href="afinsq_1.html#K5">^</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">n</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#D1" target="_self">Def1</a>, <a class="ref" href="card_fin.html#T62">CARD_FIN:62</a></i>;<br/>
<a NAME="E20:69_2"/><b>then </b>
 <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Fr2</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <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> <font color="Maroon">Fr</font></span>)</span>,<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">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T47">STIRL2_1:47</a></i>;<br/>
<a NAME="E21:69_2"/><b>then </b><i><font color="Green">E132</font></i>: 
 <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Fr2</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</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="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr</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> <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">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="binop_2.html#D9">BINOP_2:def 9</a></i>;<br/>
<a NAME="E22:69_2"/>
<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</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="E23:69_2"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">n</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="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E24:69_2"/><b>then </b><i><font color="Green">E134</font></i>: 
( <font color="Maroon">Fr1</font> <a href="catalan2.html#K5" target="_self">.</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> <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">n</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"><font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</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> 0 &amp; <span class="p1">(<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> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> 0 &amp; <font color="Maroon">Fr2</font> <a href="catalan2.html#K5" target="_self">.</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> <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">n</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"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><font color="Maroon">n</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#T4" target="_self">Th4</a>, <a class="ref" href="catalan2.html#T5" target="_self">Th5</a>, <a class="ref" href="binarith.html#T51">BINARITH:51</a>, <a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>;<br/>
<a NAME="E25:69_2"/>
(  <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Fr2</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</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">Fr2</font> <a href="catalan2.html#K5" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</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">Fr2</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><font color="Maroon">n</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"><span class="p3">(<span class="default"><font color="Maroon">n</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">n</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">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">Fr2</font> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Fr2</font> &amp; <font color="Maroon">Fr1</font> <a href="card_fin.html#K4">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <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#E8:69_2"><i><font color="Green">E62</font></i></a>, , <a class="ref" href="catalan2.html#T7" target="_self">Th7</a>, <a class="txt" href="catalan2.html#E1:69"><i><font color="Green">E51</font></i></a>, <a class="ref" href="relat_1.html#T98">RELAT_1:98</a></i>;<br/>
<a NAME="E26:69_2"/><b>then </b>
 <a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr2</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> <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">n</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><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> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="catalan2.html#E4:69_2"><i><font color="Green">E59</font></i></a>, , <a class="ref" href="catalan2.html#T6" target="_self">Th6</a>, <a class="txt" href="catalan2.html#E9"><i><font color="Green">Lemma71</font></i></a></i>;<br/>
<b>hence </b><a NAME="E27:69_2"/>
( <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> <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>  <a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr2</font> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</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> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 holds <br/><font color="Maroon">Fr2</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"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <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#T5" target="_self">Th5</a>, <a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:69"/>
for <font color="Olive">i</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">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="catalan2.html#E2:69"><i><font color="Green">E57</font></i></a>, );<br/></i>
<b>hence </b><a NAME="E4:69"/>
 ex <font color="Olive">Fr</font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <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="Olive">Fr</font> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Olive">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="Olive">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"><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> ) )
 ;<br/>


</div>
