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

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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_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="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <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="c1">n</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">tb</font> =  <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:22_1"/>
( <font color="Maroon" title="c1">n</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  or  not <font color="Maroon" title="c1">n</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1"><span class="kw">suppose </span></a><a NAME="E1:22_1_1"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<font color="Maroon" title="c1">n</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</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_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="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <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="c1">n</font></span><br/><div class="add"><a NAME="E2:22_1_1"/>
<font color="Maroon" title="c1">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> is    <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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">k</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> <span class="kw"> such that </span><br/><a NAME="E4:22_1_1"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1_1"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="abian.html#D2" title="ABIAN:def.2">ABIAN:def 2</a></span>;<br/><a NAME="E5:22_1_1"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_1_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a></span>;<br/><span class="lab"><font color="Green" title="E30">A4</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> <font color="Maroon" title="c1">n</font> = 
<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> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c3">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_1_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c3">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</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> <font color="Maroon" title="c1">n</font> = 
<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> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c3">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_1_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</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="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c3">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></span>
;<br/><span class="kw">hence </span><a NAME="E8:22_1_1"/>
<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="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <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="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_1_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E6:22_1_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="fib_num4.html#E28"><span class="lab"><font color="Green" title="E25">Lm14</font></span></a>, <a class="ref" href="power.html#T37" title="POWER:th.37">POWER:37</a></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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_2"><span class="kw">suppose </span></a><a NAME="E1:22_1_2"/>
 not <font color="Maroon" title="c1">n</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</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_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="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <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="c1">n</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">k</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> <span class="kw"> such that </span><br/><a NAME="E3:22_1_2"/><span class="lab"><font color="Green" title="E27">A5</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T9" title="ABIAN:th.9">ABIAN:9</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c4">kk</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> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span>;<br/><span class="lab"><font color="Green" title="E28">A6</font></span>: <span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> 2</span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c3">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span> <a href="square_1.html#K4" title="SQUARE_1:func.4">^2</a></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c3">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</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">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></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> 2</span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c3">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</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="square_1.html#K3" title="SQUARE_1:func.3">^2</a></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c3">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></span>
;<br/><a NAME="E6:22_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A7</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">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T34" title="POWER:th.34">POWER:34</a></span>;<br/><a NAME="E7:22_1_2"/>
<span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 3</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_1_2"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="fib_num4.html#E32"><span class="lab"><font color="Green" title="E26">Lm15</font></span></a>, <a class="ref" href="power.html#T34" title="POWER:th.34">POWER:34</a></span>;<br/><a NAME="E8:22_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 3</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="fib_num4.html#E25"><span class="lab"><font color="Green" title="E24">Lm13</font></span></a>, <a class="txt" href="fib_num4.html#E8"><span class="lab"><font color="Green" title="E8">Lm4</font></span></a>, <a class="ref" href="xreal_1.html#T69" title="XREAL_1:th.69">XREAL_1:69</a></span>;<br/><a NAME="E9:22_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span></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="complex1.html#K13" title="COMPLEX1:func.13">/</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="fib_num4.html#E9"><span class="lab"><font color="Green" title="E9">Lm5</font></span></a>, <a class="ref" href="xcmplx_1.html#T112" title="XCMPLX_1:th.112">XCMPLX_1:112</a></span>;<br/><a NAME="E10:22_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><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> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</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> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fib_num4.html#T1" target="_self" title="FIB_NUM4:th.1">Th1</a></span>;<br/><a NAME="E11:22_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><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> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</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">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T99" title="XCMPLX_1:th.99">XCMPLX_1:99</a></span>;<br/><a NAME="E12:22_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</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> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K8" title="REAL_1:func.8">*</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> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_1_2"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/><a NAME="E13:22_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><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> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span></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="complex1.html#K13" title="COMPLEX1:func.13">/</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><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> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</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">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K8" title="REAL_1:func.8">*</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> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
 ;<br/><a NAME="E14:22_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><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> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span></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="complex1.html#K13" title="COMPLEX1:func.13">/</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><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> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</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> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K8" title="REAL_1:func.8">*</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> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T99" title="XCMPLX_1:th.99">XCMPLX_1:99</a></span>;<br/><a NAME="E15:22_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><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> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span></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="complex1.html#K13" title="COMPLEX1:func.13">/</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <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">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_1_2"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>;<br/><a NAME="E16:22_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><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> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <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">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T99" title="XCMPLX_1:th.99">XCMPLX_1:99</a></span>;<br/><a NAME="E17:22_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">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 class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T69" title="XREAL_1:th.69">XREAL_1:69</a></span>;<br/><a NAME="E18:22_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><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> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">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="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">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>
 ;<br/><a NAME="E19:22_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><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> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">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#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="complex1.html#K13" title="COMPLEX1:func.13">/</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">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 class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T99" title="XCMPLX_1:th.99">XCMPLX_1:99</a></span>;<br/><a NAME="E20:22_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><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> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">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="real_1.html#K4" title="REAL_1:func.4">*</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">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 class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>;<br/><a NAME="E21:22_1_2"/><span class="kw">then </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> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">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> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span></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 NAME="E22:22_1_2"/><span class="kw">then </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> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">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> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <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"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fib_num4.html#T2" target="_self" title="FIB_NUM4:th.2">Th2</a></span>;<br/><a NAME="E23:22_1_2"/><span class="kw">then </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> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span></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> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <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"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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/><span class="kw">hence </span><a NAME="E24:22_1_2"/>
<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="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <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="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_1_2"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="ref" href="fib_num4.html#T2" target="_self" title="FIB_NUM4:th.2">Th2</a></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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
