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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span>;<br/>
<a NAME="E1:11"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<span class="p1">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> 0</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T29">POWER:29</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="fib_num.html#D1">FIB_NUM:def 1</a>, <a class="ref" href="power.html#T30">POWER:30</a></i>
<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#K2">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 5</span>)</span> <a href="xcmplx_0.html#K7">/</a> 4

<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#K2">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">5 <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 4
<b>by </b><i><a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>
<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#K2">+</a> <span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 4
<b>by </b><i><a class="ref" href="square_1.html#T97">SQUARE_1:97</a></i>
<br/>.= 
<a href="fib_num.html#K1">tau</a>  <a href="xcmplx_0.html#K3">*</a> <a href="fib_num.html#K1">tau</a> 
<b>by </b><i><a class="ref" href="fib_num.html#D1">FIB_NUM:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="fib_num.html#K1">tau</a> 
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a></i>
<br/>.= 
<a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 2</span>)</span>

;<br/>
<b>hence </b><a NAME="E2:11_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 ;<br/>


</div><b>end;</b></div>
<i><font color="Green">E28</font></i>: 1 <a href="xcmplx_0.html#K2">+</a> <a href="fib_num.html#K1">tau</a>  = 
1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 2</span>)</span>
<b>by </b><i>, <a class="ref" href="power.html#T29">POWER:29</a></i>
;<br/>
<a NAME="E3:11"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3"><b>proof </b></a><div class="add">

<span class="p1">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<a href="fib_num.html#K1">tau</a>  <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<a href="fib_num.html#K1">tau</a>  <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a></i>
<br/>.= 
<a href="fib_num.html#K1">tau</a>  <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="fib_num.html#K1">tau</a> </span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<a href="fib_num.html#K1">tau</a>  <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="xcmplx_0.html#K3">*</a> <a href="fib_num.html#K1">tau</a> </span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<a href="fib_num.html#K1">tau</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <a href="fib_num.html#K1">tau</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> 2</span>)</span>
<b>by </b><i>, <a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a></i>
;<br/>
<b>hence </b><a NAME="E2:11_3"/>
<font color="Maroon">S<sub>1</sub></font>[1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:11"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 2]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:11_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 <b>and </b><br/><a NAME="E2:11_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>

<span class="p1">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> 1</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> 2</span>)</span>
<b>by </b><i>, <a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a></i>
;<br/>
<b>hence </b><a NAME="E4:11_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:11"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="fib_num.html#S1">FIB_NUM:sch 1</a>(, , );<br/></i>
<b>hence </b><a NAME="E6:11"/>
<span class="p1">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 ;<br/>


</div>
