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

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <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> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )</span><br/>

<span class="kw">assume </span><a NAME="E1:17"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span><br/>

<a NAME="E2:17"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/>
<a NAME="E3:17"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <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="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E4:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">C1</font></span>: 
( <font color="Maroon" title="c1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_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="E1:17_1"/>
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E20">C1</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1"><span class="kw">suppose </span></a><a NAME="E1:17_1_1"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:17_1_1"/>
<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> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="17_1_2"><span class="kw">suppose </span></a><a NAME="E1:17_1_2"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span><br/><div class="add"><a NAME="E2:17_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 ;<br/><a NAME="E3:17_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">B0</font></span>: 
<font color="Maroon" title="c1">n</font> is  non  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#D1" title="NAT_2:def.1">NAT_2:def 1</a></span>;<br/><a NAME="E4:17_1_2"/>
<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> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2_1"><span class="kw">proof </span></a><div class="add">

<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><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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> $1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2;<br/>
<span class="lab"><font color="Green" title="E22">C1</font></span>: <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> </span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> 2 = 
<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> <a href="power.html#K3" title="POWER:func.3">to_power</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
<br/>.= 
<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> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T53" title="POWER:th.53">POWER:53</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <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> <span class="p4">(<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="real_1.html#K7" title="REAL_1:func.7">+</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="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</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">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 5</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 4
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default">3 <a href="real_1.html#K9" title="REAL_1:func.9">-</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="real_1.html#K10" title="REAL_1:func.10">/</a> 2

;<br/>
<a NAME="E2:17_1_2_1"/>
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T85" title="SQUARE_1:th.85">SQUARE_1:85</a>, <a class="ref" href="square_1.html#T95" title="SQUARE_1:th.95">SQUARE_1:95</a></span>;<br/>
<a NAME="E3:17_1_2_1"/><span class="kw">then </span>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></span>;<br/>
<a NAME="E4:17_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</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="real_1.html#K7" title="REAL_1:func.7">+</a> 3 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/>
<a NAME="E5:17_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">B1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[2]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1_2_1"><span class="lab"><font color="Green" title="E22">C1</font></span></a>, <a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></span>;<br/>
<a NAME="E6:17_1_2_1"/><span class="lab"><font color="Green" title="E24">B2</font></span>: 
for <font color="Olive" title="b1">k</font> being non  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  <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>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">k</font> be  non  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  <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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:17_1_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]</span><br/>

<a NAME="E2:17_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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> <font color="Maroon" title="c2">k</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/>
<a NAME="E3:17_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">b3</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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> <font color="Maroon" title="c2">k</font></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="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> </span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></span>;<br/>
<a NAME="E4:17_1_2_1_2"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fib_num4.html#T5" target="_self" title="FIB_NUM4:th.5">hop3</a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></span>;<br/>
<a NAME="E5:17_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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> <font color="Maroon" title="c2">k</font></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="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> </span>)</span> <a href="power.html#K4" title="POWER:func.4">to_power</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <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="#E3:17_1_2_1_2"><span class="lab"><font color="Green" title="E25">b3</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:17_1_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fib_num4.html#T2" target="_self" title="FIB_NUM4:th.2">JakPower32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:17_1_2_1"/><span class="lab"><font color="Green" title="E25">bb4</font></span>: 
for <font color="Olive" title="b1">n</font> being non  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  <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">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_2.html#S2" title="NAT_2:sch.2">NAT_2:sch 2</a>(<a class="txt" href="#E5:17_1_2_1"><span class="lab"><font color="Green" title="E23">B1</font></span></a>, <a class="txt" href="#E6:17_1_2_1"><span class="lab"><font color="Green" title="E24">B2</font></span></a>);<br/></span>
<a NAME="E8:17_1_2_1"/>
for <font color="Olive" title="b1">n</font> being non  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <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="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">n</font> be  non  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  <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="Red">::  thesis: </font></span></a><span class="hide"> <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="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span><br/>

<a NAME="E1:17_1_2_1_3"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <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="#E7:17_1_2_1"><span class="lab"><font color="Green" title="E25">bb4</font></span></a></span>;<br/>
<a NAME="E2:17_1_2_1_3"/><span class="kw">then </span>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="c2">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &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> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</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="c2">n</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#T11" title="ABSVALUE:th.11">ABSVALUE:11</a>, <a class="ref" href="series_1.html#T2" title="SERIES_1:th.2">SERIES_1:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:17_1_2_1_3"/>
<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="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:17_1_2_1"/>
<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> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <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="#E3:17_1_2"><span class="lab"><font color="Green" title="E21">B0</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:17_1_2"/>
<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> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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>
