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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means  <a href="pre_ff.html#K3">Fib</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K1" target="_self">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span>;<br/>
<a NAME="E1:21"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<a NAME="E1:21_1"/>
<a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="power.html#K3">to_power</a> 0 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="power.html#T29">POWER:29</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K1" target="_self">tau</a>  <a href="power.html#K3">to_power</a> 0</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="power.html#K3">to_power</a> 0</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span> = 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T29">POWER:29</a></i>
<br/>.= 
0

;<br/>
<b>hence </b><a NAME="E3:21_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i><a class="ref" href="pre_ff.html#T1">PRE_FF:1</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">c<sub>1</sub></font> = 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> = 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span>;<br/>
<a NAME="E2:21"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<a NAME="E1:21_2"/><i><font color="Green">E23</font></i>: 
<a href="fib_num.html#K1" target="_self">tau</a>  <a href="xcmplx_0.html#K6">-</a> <a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> 5
 
;<br/>
<a NAME="E2:21_2"/>
<a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="power.html#K3">to_power</a> 1 <a href="hidden.html#R1">=</a>  <a href="fib_num.html#K2" target="_self">tau_bar</a> 
 
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K1" target="_self">tau</a>  <a href="power.html#K3">to_power</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="power.html#K3">to_power</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span> = 
<span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span>
<b>by </b><i><a class="txt" href="fib_num.html#E1:21_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
 <a href="pre_ff.html#K3">Fib</a> 1
<b>by </b><i><a class="txt" href="fib_num.html#E19"><i><font color="Green">Lemma15</font></i></a>, <a class="ref" href="pre_ff.html#T1">PRE_FF:1</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
;<br/>
<b>hence </b><a NAME="E4:21_2"/>
<font color="Maroon">S<sub>1</sub></font>[1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:21"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 2]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:21_3"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 <b>and </b><br/><a NAME="E2:21_3"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> = <a href="fib_num.html#K1" target="_self">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <a href="fib_num.html#K1" target="_self">tau</a>  <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <a href="fib_num.html#K1" target="_self">tau</a>  <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span>;<br/>
<i><font color="Green">E26</font></i>: <a href="fib_num.html#K1" target="_self">tau</a>  <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" target="_self">tau</a>  <a href="power.html#K3">to_power</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="fib_num.html#K1" target="_self">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fib_num.html#E22"><i><font color="Green">Lemma17</font></i></a>, <a class="ref" href="power.html#T32">POWER:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" target="_self">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="fib_num.html#K1" target="_self">tau</a>  <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="fib_num.html#E17"><i><font color="Green">Lemma13</font></i></a>, <a class="ref" href="power.html#T53">POWER:53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K1" target="_self">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="fib_num.html#K1" target="_self">tau</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K1" target="_self">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K1" target="_self">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="fib_num.html#K1" target="_self">tau</a>  <a href="power.html#K3">to_power</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="fib_num.html#K1" target="_self">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" target="_self">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="fib_num.html#K1" target="_self">tau</a>  <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fib_num.html#E22"><i><font color="Green">Lemma17</font></i></a>, <a class="ref" href="power.html#T32">POWER:32</a></i>
;<br/>
<a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> = 
<a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
<b>by </b><i><a class="txt" href="fib_num.html#E23"><i><font color="Green">Lemma18</font></i></a>, <a class="ref" href="power.html#T46">POWER:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="newton.html#K2">|^</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="xcmplx_0.html#K3">*</a> <a href="fib_num.html#K2" target="_self">tau_bar</a> </span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="fib_num.html#E17"><i><font color="Green">Lemma13</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="fib_num.html#K2" target="_self">tau_bar</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="fib_num.html#E23"><i><font color="Green">Lemma18</font></i></a>, <a class="ref" href="power.html#T46">POWER:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fib_num.html#E23"><i><font color="Green">Lemma18</font></i></a>, <a class="ref" href="power.html#T46">POWER:46</a></i>
;<br/>
<a NAME="E5:21_3"/><b>then </b><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K1" target="_self">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K1" target="_self">tau</a>  <a href="power.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="power.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fib_num.html#K1" target="_self">tau</a>  <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="fib_num.html#E3:21_3"><i><font color="Green">E26</font></i></a></i>;<br/>
 <a href="pre_ff.html#K3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> = 
 <a href="pre_ff.html#K3">Fib</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fib_num.html#K1" target="_self">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fib_num.html#K1" target="_self">tau</a>  <a href="power.html#K3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="power.html#K3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fib_num.html#E1:21_3"><i><font color="Green">E24</font></i></a>, <a class="txt" href="fib_num.html#E2:21_3"><i><font color="Green">E25</font></i></a>, <a class="ref" href="pre_ff.html#T1">PRE_FF:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K1" target="_self">tau</a>  <a href="power.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="power.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span>
<b>by </b><i><a class="txt" href="fib_num.html#E5:21_3"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xcmplx_1.html#T63">XCMPLX_1:63</a></i>
;<br/>
<b>hence </b><a NAME="E7:21_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 2]
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:21"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="fib_num.html#S1" target="_self">FIB_NUM:sch 1</a>(<a class="txt" href="fib_num.html#E1:21"><i><font color="Green">E21</font></i></a>, <a class="txt" href="fib_num.html#E2:21"><i><font color="Green">E22</font></i></a>, <a class="txt" href="fib_num.html#E3:21"><i><font color="Green">E23</font></i></a>);</i><br/>


</div>
