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

<b>let </b><font color="Maroon" title="c1">seq1</font>, <font color="Maroon" title="c2">seq2</font>, <font color="Maroon" title="c3">seq3</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">seq1</font> <a href="catalan2.html#K5" title="CATALAN2:func.5">(##)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c3">seq3</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="catalan2.html#K5" title="CATALAN2:func.5">(##)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="catalan2.html#K5" title="CATALAN2:func.5">(##)</a> <font color="Maroon" title="c3">seq3</font></span>)</span></span><br/>





<b>set </b><font color="Maroon" title="c4">S</font> = <font color="Maroon" title="c2">seq2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c3">seq3</font>;<br/>
<b>set </b><font color="Maroon" title="c5">S2</font> = <font color="Maroon" title="c1">seq1</font> <a href="catalan2.html#K5" title="CATALAN2:func.5">(##)</a> <font color="Maroon" title="c2">seq2</font>;<br/>
<b>set </b><font color="Maroon" title="c6">S3</font> = <font color="Maroon" title="c1">seq1</font> <a href="catalan2.html#K5" title="CATALAN2:func.5">(##)</a> <font color="Maroon" title="c3">seq3</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="catalan2.html#K5" title="CATALAN2:func.5">(##)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c3">seq3</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="catalan2.html#K5" title="CATALAN2:func.5">(##)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="catalan2.html#K5" title="CATALAN2:func.5">(##)</a> <font color="Maroon" title="c3">seq3</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font> )</span><br/><b>assume </b><a NAME="E1:66_1"/><i><font color="Green" title="E54">A1</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="catalan2.html#K5" title="CATALAN2:func.5">(##)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c3">seq3</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="catalan2.html#K5" title="CATALAN2:func.5">(##)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="catalan2.html#K5" title="CATALAN2:func.5">(##)</a> <font color="Maroon" title="c3">seq3</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/><b>reconsider </b><font color="Maroon" title="c8">k</font> = <font color="Maroon" title="c7">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E1:66_1"><i><font color="Green" title="E54">A1</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c9">Fr</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E4:66_1"/><i><font color="Green" title="E55">A2</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">Fr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>and </b><br/><a NAME="E5:66_1"/><i><font color="Green" title="E56">A3</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/><font color="Maroon" title="c9">Fr</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c3">seq3</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E6:66_1"/><i><font color="Green" title="E57">A4</font></i>: 
 <a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <font color="Maroon" title="c9">Fr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="catalan2.html#K5" title="CATALAN2:func.5">(##)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c3">seq3</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="ref" href="catalan2.html#D5" target="_self" title="CATALAN2:def.5">Def5</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">Fr1</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E8:66_1"/><i><font color="Green" title="E58">A5</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c10">Fr1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>and </b><br/><a NAME="E9:66_1"/><i><font color="Green" title="E59">A6</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/><font color="Maroon" title="c10">Fr1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E10:66_1"/><i><font color="Green" title="E60">A7</font></i>: 
 <a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <font color="Maroon" title="c10">Fr1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="catalan2.html#K5" title="CATALAN2:func.5">(##)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="ref" href="catalan2.html#D5" target="_self" title="CATALAN2:def.5">Def5</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">Fr2</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E12:66_1"/><i><font color="Green" title="E61">A8</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c11">Fr2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>and </b><br/><a NAME="E13:66_1"/><i><font color="Green" title="E62">A9</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/><font color="Maroon" title="c11">Fr2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E14:66_1"/><i><font color="Green" title="E63">A10</font></i>: 
 <a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <font color="Maroon" title="c11">Fr2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="catalan2.html#K5" title="CATALAN2:func.5">(##)</a> <font color="Maroon" title="c3">seq3</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="ref" href="catalan2.html#D5" target="_self" title="CATALAN2:def.5">Def5</a></i>;<br/><a NAME="E15:66_1"/><i><font color="Green" title="E64">A11</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">Fr</font> holds <br/><font color="Maroon" title="c9">Fr</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Fr1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">Fr2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">Fr</font> implies <font color="Maroon" title="c9">Fr</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Fr1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">Fr2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:66_1_1"/><i><font color="Green" title="E65">A12</font></i>: 
<font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">Fr</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">Fr</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Fr1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">Fr2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span></span><br/>

<a NAME="E2:66_1_1"/>
( <font color="Maroon" title="c10">Fr1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c11">Fr2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c9">Fr</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c3">seq3</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c3">seq3</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E4:66_1"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="#E5:66_1"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="#E9:66_1"><i><font color="Green" title="E59">A6</font></i></a>, <a class="txt" href="#E13:66_1"><i><font color="Green" title="E62">A9</font></i></a>, <a class="txt" href="#E1:66_1_1"><i><font color="Green" title="E65">A12</font></i></a>, <a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></i>;<br/>
<a NAME="E3:66_1_1"/><b>then </b>
<font color="Maroon" title="c9">Fr</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Fr1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">Fr2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E4:66_1_1"/>
<font color="Maroon" title="c9">Fr</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Fr1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">Fr2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="binop_2.html#D9" title="BINOP_2:def.9">BINOP_2:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E16:66_1"/>
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c10">Fr1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">Fr2</font> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c10">Fr1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c9">Fr</font> )
 <b>by </b><i><a class="txt" href="#E4:66_1"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="#E8:66_1"><i><font color="Green" title="E58">A5</font></i></a>, <a class="txt" href="#E12:66_1"><i><font color="Green" title="E61">A8</font></i></a></i>;<br/><a NAME="E17:66_1"/><b>then </b>
<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Fr1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c11">Fr2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c9">Fr</font>
 <b>by </b><i><a class="txt" href="#E15:66_1"><i><font color="Green" title="E64">A11</font></i></a>, <a class="ref" href="card_fin.html#T62" title="CARD_FIN:th.62">CARD_FIN:62</a></i>;<br/><a NAME="E18:66_1"/><b>then </b>
 <a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <font color="Maroon" title="c9">Fr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <font color="Maroon" title="c10">Fr1</font></span>)</span>,<span class="p1">(<span class="default"><a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <font color="Maroon" title="c11">Fr2</font></span>)</span>
 <b>by </b><i><a class="ref" href="stirl2_1.html#T47" title="STIRL2_1:th.47">STIRL2_1:47</a></i>;<br/><a NAME="E19:66_1"/><b>then </b>
 <a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <font color="Maroon" title="c9">Fr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <font color="Maroon" title="c10">Fr1</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" title="CATALAN2:func.4">Sum</a> <font color="Maroon" title="c11">Fr2</font></span>)</span>
 <b>by </b><i><a class="ref" href="binop_2.html#D9" title="BINOP_2:def.9">BINOP_2:def 9</a></i>;<br/><b>hence </b><a NAME="E20:66_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="catalan2.html#K5" title="CATALAN2:func.5">(##)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c3">seq3</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="catalan2.html#K5" title="CATALAN2:func.5">(##)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="catalan2.html#K5" title="CATALAN2:func.5">(##)</a> <font color="Maroon" title="c3">seq3</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 <b>by </b><i><a class="txt" href="#E6:66_1"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="#E10:66_1"><i><font color="Green" title="E60">A7</font></i></a>, <a class="txt" href="#E14:66_1"><i><font color="Green" title="E63">A10</font></i></a>, <a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:66"/>
<font color="Maroon" title="c1">seq1</font> <a href="catalan2.html#K5" title="CATALAN2:func.5">(##)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq2</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c3">seq3</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="catalan2.html#K5" title="CATALAN2:func.5">(##)</a> <font color="Maroon" title="c2">seq2</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq1</font> <a href="catalan2.html#K5" title="CATALAN2:func.5">(##)</a> <font color="Maroon" title="c3">seq3</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
