<?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">seq4</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:64_1_2"/><i><font color="Green">E59</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> )
 <b>and </b><br/><a NAME="E2:64_1_2"/><i><font color="Green">E60</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">seq4</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:64_1_2_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">x</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="catalan2.html#T2" target="_self">Th2</a></i>;<br/><b>consider </b><font color="Maroon">Fr1</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:64_1_2_1"/><i><font color="Green">E62</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="binop_2.html#K23">+</a> 1
 <b>and </b><br/><a NAME="E5:64_1_2_1"/><i><font color="Green">E93</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="binop_2.html#K23">+</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="E6:64_1_2_1"/><i><font color="Green">E95</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="E8:64_1_2_1"/><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> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 <b>and </b><br/><a NAME="E9:64_1_2_1"/><i><font color="Green">E97</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</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">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">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="E10:64_1_2_1"/><i><font color="Green">E98</font></i>: 
 <a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq4</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:64_1_2_1_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr1</font>
 ;<br/><a NAME="E2:64_1_2_1_1"/>
( <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; <font color="Maroon">Fr2</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> )
 <b>by </b><i><a class="ref" href="catalan2.html#T3" target="_self">Th3</a>, , <a class="txt" href="catalan2.html#E2:64_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="catalan2.html#E4:64_1_2_1"><i><font color="Green">E62</font></i></a></i>;<br/><b>hence </b><a NAME="E3:64_1_2_1_1"/>
<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> <font color="Maroon">n</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:64_1_2_1"/>
<font color="Maroon">seq3</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq4</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="catalan2.html#T3" target="_self">Th3</a>, , <a class="txt" href="catalan2.html#E1:64_1_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="catalan2.html#E1:64_1_2_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="afinsq_1.html#T10">AFINSQ_1:10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:64_1_2"/>
<font color="Maroon">seq3</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">seq4</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T8">SEQ_1:8</a></i>;<br/>


</div>
