<?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">seq3</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>





<b>set </b><font color="Maroon">S</font> = <font color="Maroon">seq2</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">seq3</font>;<br/>
<b>set </b><font color="Maroon">S2</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">S3</font> = <font color="Maroon">seq1</font> <a href="catalan2.html#K6" target="_self">(##)</a> <font color="Maroon">seq3</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_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="E1:67_1"/><i><font color="Green">E51</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">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="E4:67_1"/><i><font color="Green">E53</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr</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="E5:67_1"/><i><font color="Green">E54</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">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"><span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">seq3</font></span>)</span> <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="E6:67_1"/><i><font color="Green">E57</font></i>: 
 <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> <span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">seq3</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="catalan2.html#E4:67_1"><i><font color="Green">E53</font></i></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="E8:67_1"/><i><font color="Green">E58</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="E9:67_1"/><i><font color="Green">E59</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="E10:67_1"/><i><font color="Green">E60</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> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="catalan2.html#E4:67_1"><i><font color="Green">E53</font></i></a></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="E12:67_1"/><i><font color="Green">E61</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="E13:67_1"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="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">seq3</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="E14:67_1"/><i><font color="Green">E93</font></i>: 
 <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"><font color="Maroon">seq1</font> <a href="catalan2.html#K6" target="_self">(##)</a> <font color="Maroon">seq3</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="catalan2.html#E4:67_1"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E15:67_1"/><i><font color="Green">E95</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr</font> holds <br/><font color="Maroon">Fr</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Olive">n</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">Fr1</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Olive">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">Fr2</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Olive">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1"><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>assume </b><a NAME="E1:67_1_1"/><i><font color="Green">E96</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">Fr</font>
 ;<br/>

<a NAME="E2:67_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">seq3</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">Fr</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"><span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">seq3</font></span>)</span> <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; <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">seq3</font></span>)</span> <a href="seq_1.html#K2">.</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> <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#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">seq3</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#E6:67_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="catalan2.html#E10:67_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="catalan2.html#T4" target="_self">Th4</a>, , <a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/>
<a NAME="E3:67_1_1"/><b>then </b>
<font color="Maroon">Fr</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">Fr1</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">Fr2</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E4:67_1_1"/>
<font color="Maroon">Fr</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">n</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">Fr1</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">Fr2</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">n</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><a NAME="E16:67_1"/>
 <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="E17:67_1"/><b>then </b>
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font> <a href="hidden.html#R1">=</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">Fr1</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="ref" href="catalan2.html#T3" target="_self">Th3</a>, <a class="txt" href="catalan2.html#E9:67_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="catalan2.html#E13:67_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E18:67_1"/><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">Fr1</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">Fr2</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K33">addreal</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">Fr</font>
 <b>by </b><i>, <a class="ref" href="card_fin.html#T62">CARD_FIN:62</a></i>;<br/><a NAME="E19:67_1"/><b>then </b>
 <a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr</font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr1</font></span>)</span>,<span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr2</font></span>)</span>
 <b>by </b><i><a class="ref" href="stirl2_1.html#T47">STIRL2_1:47</a></i>;<br/><a NAME="E20:67_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"><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">Fr2</font></span>)</span>
 <b>by </b><i><a class="ref" href="binop_2.html#D9">BINOP_2:def 9</a></i>;<br/><b>hence </b><a NAME="E21:67_1"/>
<span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="catalan2.html#K6" target="_self">(##)</a> <span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">seq3</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</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#K9">+</a> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="catalan2.html#K6" target="_self">(##)</a> <font color="Maroon">seq3</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a>, <a class="txt" href="catalan2.html#E8:67_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="catalan2.html#E12:67_1"><i><font color="Green">E61</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:67"/>
<font color="Maroon">seq1</font> <a href="catalan2.html#K6" target="_self">(##)</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">seq3</font></span>)</span> <a href="funct_2.html#R2">=</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#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="catalan2.html#K6" target="_self">(##)</a> <font color="Maroon">seq3</font></span>)</span>
 <b>by </b><i><a class="ref" href="seq_1.html#T8">SEQ_1:8</a></i>;<br/>


</div>
