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

<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="fib_num3.html#K3" title="FIB_NUM3:func.3">GenFib</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> <a href="fib_num3.html#K3" title="FIB_NUM3:func.3">GenFib</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,$1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> $1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> $1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span>

<br/>.= 
<font color="Maroon" title="c2">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fib_num.html#D1" title="FIB_NUM:def.1">FIB_NUM:def 1</a>, <a class="ref" href="fib_num.html#D2" title="FIB_NUM:def.2">FIB_NUM:def 2</a>, <a class="ref" href="xcmplx_1.html#T89" title="XCMPLX_1:th.89">XCMPLX_1:89</a></span>
<br/>.= 
 <a href="fib_num3.html#K3" title="FIB_NUM3:func.3">GenFib</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,1)
<span class="kw">by</span> <span class="lab"><a class="ref" href="fib_num3.html#T32" target="_self" title="FIB_NUM3:th.32">Th32</a></span>
;<br/>
<a NAME="E2:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
;<br/>
<a NAME="E3:59"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2] )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">c</font> = <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">d</font> = <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font>;<br/>
<span class="lab"><font color="Green" title="E32">A3</font></span>: <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T27" title="POWER:th.27">POWER:27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>
;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">g</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span>;<br/>
<span class="lab"><font color="Green" title="E33">A4</font></span>: <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fib_num2.html#T5" title="FIB_NUM2:th.5">FIB_NUM2:5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>
;<br/>
<a NAME="E3:59_2"/>
<span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fib_num3.html#T2" target="_self" title="FIB_NUM3:th.2">Th2</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E34">A5</font></span>: 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fib_num.html#D1" title="FIB_NUM:def.1">FIB_NUM:def 1</a></span>
<br/>.= 
<a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fib_num.html#D1" title="FIB_NUM:def.1">FIB_NUM:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>
<br/>.= 
<a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T27" title="POWER:th.27">POWER:27</a></span>
;<br/>
<span class="lab"><font color="Green" title="E35">A6</font></span>: 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 5</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4
<span class="kw">by</span> <span class="lab"><a class="ref" href="fib_num.html#D2" title="FIB_NUM:def.2">FIB_NUM:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4
<span class="kw">by</span> <span class="lab"><a class="ref" href="fib_num3.html#T2" target="_self" title="FIB_NUM3:th.2">Th2</a></span>
<br/>.= 
<a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fib_num.html#D2" title="FIB_NUM:def.2">FIB_NUM:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>
<br/>.= 
<a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fib_num2.html#T5" title="FIB_NUM2:th.5">FIB_NUM2:5</a></span>
<br/>.= 
<a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> 2

;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">h</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span>;<br/>
<a NAME="E6:59_2"/><span class="lab"><font color="Green" title="E36">A7</font></span>: 
( <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> 2</span>)</span> &amp; <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> 2</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fib_num2.html#T5" title="FIB_NUM2:th.5">FIB_NUM2:5</a></span>;<br/>
<span class="kw">assume </span><a NAME="E7:59_2"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2]</span><br/>

<span class="kw">then </span> <a href="fib_num3.html#K3" title="FIB_NUM3:func.3">GenFib</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>) = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_2"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E2:59_2"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="ref" href="fib_num3.html#T34" target="_self" title="FIB_NUM3:th.34">Th34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p0">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T62" title="XCMPLX_1:th.62">XCMPLX_1:62</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:59_2"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="txt" href="#E4:59_2"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E5:59_2"><span class="lab"><font color="Green" title="E35">A6</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:59_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T24" title="POWER:th.24">POWER:24</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T24" title="POWER:th.24">POWER:24</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span>

<br/>.= 
<font color="Maroon" title="c1">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fib_num.html#D1" title="FIB_NUM:def.1">FIB_NUM:def 1</a>, <a class="ref" href="fib_num.html#D2" title="FIB_NUM:def.2">FIB_NUM:def 2</a>, <a class="ref" href="xcmplx_1.html#T89" title="XCMPLX_1:th.89">XCMPLX_1:89</a></span>
<br/>.= 
 <a href="fib_num3.html#K3" title="FIB_NUM3:func.3">GenFib</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="fib_num3.html#T32" target="_self" title="FIB_NUM3:th.32">Th32</a></span>
;<br/>
<a NAME="E5:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A8</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E6:59"/>
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="fib_num.html#S1" title="FIB_NUM:sch.1">FIB_NUM:sch 1</a>(<a class="txt" href="#E5:59"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E31">A2</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E7:59"/>
 <a href="fib_num3.html#K3" title="FIB_NUM3:func.3">GenFib</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
