<?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_num3.html#K1" target="_self">Lucas</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</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>;<br/>
<a NAME="E1:28"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i>, <a class="ref" href="pre_ff.html#T1">PRE_FF:1</a></i>;<br/>
<a NAME="E2:28"/>
<span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 2
 
;<br/>
<a NAME="E3:28"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="pre_ff.html#K3">Fib</a> 2 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="pre_ff.html#T1">PRE_FF:1</a></i>;<br/>
<a NAME="E4:28"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<b>by </b><i>, , <a class="ref" href="pre_ff.html#T1">PRE_FF:1</a></i>;<br/>
<a NAME="E5:28"/><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="28_1"><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="E1:28_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 <b>and </b><br/><a NAME="E2:28_1"/><i><font color="Green">E51</font></i>: 
<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_num3.html#K1" target="_self">Lucas</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="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num3.html#K1" target="_self">Lucas</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="fib_num3.html#K1" target="_self">Lucas</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num3.html#K1" target="_self">Lucas</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="fib_num3.html#K1" target="_self">Lucas</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fib_num2.html#T26">FIB_NUM2:26</a></i>
<br/>.= 
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fib_num3.html#E1:28"><i><font color="Green">E27</font></i></a>, <a class="txt" href="fib_num3.html#E3:28"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span>

<br/>.= 
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fib_num2.html#T27">FIB_NUM2:27</a></i>
<br/>.= 
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</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>

;<br/>
<b>hence </b><a NAME="E4:28_1"/>
<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="E6:28"/>
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>(, , <a class="ref" href="fib_num3.html#T1" target="_self">Th1</a>);<br/></i>
<b>hence </b><a NAME="E7:28"/>
<span class="p1">(<span class="default"><a href="fib_num3.html#K1" target="_self">Lucas</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/>


</div>
