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

<b>let </b><font color="Maroon">Fr1</font> be   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">Fr2</font> be   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:63"/><i><font color="Green">E51</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr1</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr2</font>
 <b>and </b><br/><a NAME="E2:63"/><i><font color="Green">E53</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#K1">len</a> <font color="Maroon">Fr1</font> 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> <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="Olive">n</font></span>)</span></span>)</span>
 ;<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"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Olive">i</font></span>)</span>;<br/>
<a NAME="E3:63"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </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="E1:63_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font>
 ;<br/>

<a NAME="E2:63_1"/>
 <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T8">STIRL2_1:8</a></i>;<br/>
<b>then 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#T3" target="_self">Th3</a></i>;<br/>
<b>take </b>
<span class="p1">(<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="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span>
;<br/>

<a NAME="E4:63_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font> &amp; 1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="catalan2.html#T3" target="_self">Th3</a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E5:63_1"/><b>then </b>
( <font color="Maroon">k</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">Fr1</font> &amp; <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font></span>)</span> <a href="binop_2.html#K23">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E6:63_1"/><b>then </b>
( <span class="p1">(<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="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</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="binop_2.html#K23">+</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<b>hence </b><a NAME="E7:63_1"/>
( <span class="p1">(<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="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<span class="p1">(<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="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span>] )
 <b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">P</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font>, <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font><b> such that </b><br/><a NAME="E5:63"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">P</font> <a href="stirl2_1.html#K2">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="ref" href="catalan2.html#T2" target="_self">Th2</a>);<br/></i>
<a NAME="E6:63"/>
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font> &amp; <font color="Maroon">P</font> <a href="stirl2_1.html#K2">.</a> <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="stirl2_1.html#K2">.</a> <font color="Olive">x2</font> holds <br/><font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:63_2"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font> &amp; <font color="Maroon">P</font> <a href="stirl2_1.html#K2">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="stirl2_1.html#K2">.</a> <font color="Maroon">x2</font> )
 ;<br/>

<a NAME="E2:63_2"/>
 <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T8">STIRL2_1:8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">x1</font>, <font color="Maroon">j</font> = <font color="Maroon">x2</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E4:63_2"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font> )
 
<b>by </b><i>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E5:63_2"/><b>then </b>
( <font color="Maroon">i</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">Fr1</font> &amp; <font color="Maroon">j</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">Fr1</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:63_2"/><b>then </b>
( <span class="p1">(<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="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">i</font></span>)</span> &amp; <span class="p1">(<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="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E7:63_2"/><b>then </b>
( <font color="Maroon">P</font> <a href="stirl2_1.html#K2">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">P</font> <a href="stirl2_1.html#K2">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="catalan2.html#T3" target="_self">Th3</a>, </i>;<br/>
<b>hence </b><a NAME="E8:63_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E7:63"/><b>then </b>
( <font color="Maroon">P</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#T77">FUNCT_2:77</a></i>;<br/>
<a NAME="E8:63"/><b>then </b>
( <font color="Maroon">P</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">P</font> is <a href="funct_2.html#V2">onto</a> )
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T70">STIRL2_1:70</a></i>;<br/>
<a NAME="E9:63"/><b>then </b>
( <font color="Maroon">P</font> is <a href="funct_2.html#V3">bijective</a> &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="funct_2.html#D4">FUNCT_2:def 4</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr1</font> ;<br/>
<div><i><font color="Green">E60</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3"><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="E1:63_3"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr1</font>
 ;<br/><a NAME="E2:63_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="catalan2.html#E5:63"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funct_2.html#T67">FUNCT_2:67</a></i>;<br/><a NAME="E3:63_3"/><b>then </b>
<font color="Maroon">P</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E4:63_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr2</font> )
 <b>by </b><i><a class="txt" href="catalan2.html#E2:63"><i><font color="Green">E53</font></i></a>, <a class="txt" href="catalan2.html#E5:63"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funct_2.html#T67">FUNCT_2:67</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:63"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr2</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr1</font>
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_4"><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="E1:63_4"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr1</font>
 ;<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/>;<br/><a NAME="E3:63_4"/>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr1</font> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr1</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</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="E4:63_4"/><b>then </b>
( <font color="Maroon">P</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<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="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">Fr1</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">k</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">k</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="catalan2.html#E3:63"><i><font color="Green">E54</font></i></a>, <a class="ref" href="catalan2.html#T3" target="_self">Th3</a></i>;<br/><b>hence </b><a NAME="E5:63_4"/>
<font color="Maroon">Fr1</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">x</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"><font color="Maroon">P</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<a NAME="E14:63"/><b>then </b>
<font color="Maroon">Fr1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fr2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="txt" href="catalan2.html#E5:63"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funct_1.html#T20">FUNCT_1:20</a></i>;<br/>
<b>hence </b><a NAME="E15:63"/>
 <a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr1</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="txt" href="catalan2.html#E2:63"><i><font color="Green">E53</font></i></a>, <a class="ref" href="card_fin.html#T48">CARD_FIN:48</a></i>;<br/>


</div>
