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

<b>let </b><font color="Maroon">seq3</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><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><a NAME="E1:64_1_3"/><i><font color="Green">E129</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  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/>(  <a href="afinsq_1.html#K2">dom</a> <font color="Olive">Fr</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; ( 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="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Olive">k</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">n</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">n</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"><font color="Olive">k</font> <a href="binarith.html#K3">-'</a> <font color="Olive">n</font></span>)</span></span>)</span> ) &amp;  <a href="catalan2.html#K4" target="_self">Sum</a> <font color="Olive">Fr</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq3</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> )
 ;<br/>

<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>reconsider </b><font color="Maroon">k'</font> = <font color="Maroon">k</font> 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="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<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">seq2</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">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Olive">i</font></span>)</span></span>)</span>;<br/>
<a NAME="E3:64_1_3"/><i><font color="Green">E131</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">k'</font> <a href="binop_2.html#K23">+</a> 1 holds <br/> ex <font color="Olive">z</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">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_3_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="E1:64_1_3_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k'</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/>

<b>take </b><font color="Maroon">S</font> = <span class="p1">(<span class="default"><font color="Maroon">seq2</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">seq1</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span></span>)</span>;<br/>

<b>thus </b><a NAME="E2:64_1_3_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">S</font>]
 ;<br/>


</div><b>end;</b></div>
<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="E5:64_1_3"/><i><font color="Green">E132</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>and </b><br/><a NAME="E6:64_1_3"/><i><font color="Green">E133</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="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 holds <br/><font color="Maroon">Fr1</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Olive">n</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">n</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"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Olive">n</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E7:64_1_3"/><i><font color="Green">E134</font></i>: 
 <a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/>
<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="E9:64_1_3"/><i><font color="Green">E135</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k'</font> <a href="binop_2.html#K23">+</a> 1
 <b>and </b><br/><a NAME="E10:64_1_3"/><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">k'</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">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/>

<b>thus </b><a NAME="E11:64_1_3"/>
(  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; ( 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="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <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">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="binop_2.html#K11">*</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="binarith.html#K3">-'</a> <font color="Olive">n</font></span>)</span></span>)</span> ) )
 <b>by </b><i>, </i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_3_2"><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/><b>assume </b><a NAME="E1:64_1_3_2"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font>
 ;<br/><a NAME="E2:64_1_3_2"/><i><font color="Green">E140</font></i>: 
 <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="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E3:64_1_3_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="catalan2.html#T2" target="_self">Th2</a>, , <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/><a NAME="E4:64_1_3_2"/><b>then </b><i><font color="Green">E141</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:64_1_3_2"/><b>then </b>
( <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr2</font> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr2</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr2</font> &amp; <font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 <b>by </b><i>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="xreal_1.html#T45">XREAL_1:45</a></i>;<br/><a NAME="E6:64_1_3_2"/><b>then </b><i><font color="Green">E142</font></i>: 
( <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr2</font></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> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> &amp; <font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E7:64_1_3_2"/><b>then </b><i><font color="Green">E144</font></i>: 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr2</font></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#R2">in</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/><a NAME="E8:64_1_3_2"/>
( <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">n</font> &amp; <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E9:64_1_3_2"/><b>then </b>
( <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E10:64_1_3_2"/><b>then </b>
( <font color="Maroon">Fr2</font> <a href="catalan2.html#K5" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr2</font></span>)</span> <a href="binarith.html#K3">-'</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"><font color="Maroon">seq2</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</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> &amp; <font color="Maroon">Fr1</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">n</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">n</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"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span>)</span> &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="catalan2.html#T2" target="_self">Th2</a>, <a class="ref" href="catalan2.html#T3" target="_self">Th3</a>, , , , <a class="ref" href="catalan2.html#T4" target="_self">Th4</a>, </i>;<br/><b>hence </b><a NAME="E11:64_1_3_2"/>
<font color="Maroon">Fr1</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fr2</font> <a href="catalan2.html#K5" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="catalan2.html#T2" target="_self">Th2</a>, , <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:64_1_3"/>
<font color="Maroon">seq3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr2</font>
 <b>by </b><i><a class="ref" href="catalan2.html#T2" target="_self">Th2</a>, , , </i>;<br/>


</div>
