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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means  <a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,0</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <font color="Maroon">a<sub>1</sub></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> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E1:50"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="fib_num3.html#T11" target="_self">Th11</a>, <a class="ref" href="pre_ff.html#T1">PRE_FF:1</a></i>;<br/>
<a NAME="E2:50"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<a NAME="E1:50_1"/>
 <a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="fib_num3.html#T32" target="_self">Th32</a></i>;<br/>
<b>then </b> <a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,0</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> 2
<b>by </b><i><a class="ref" href="fib_num3.html#T32" target="_self">Th32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,0</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_ff.html#K3">Fib</a> 1</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> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fib_num3.html#T11" target="_self">Th11</a>, <a class="ref" href="pre_ff.html#T1">PRE_FF:1</a></i>
;<br/>
<b>hence </b><a NAME="E3:50_1"/>
<font color="Maroon">S<sub>1</sub></font>[1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:50"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 2]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:50_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 <b>and </b><br/><a NAME="E2:50_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>

 <a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,0</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="fib_num3.html#K1" target="_self">Lucas</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fib_num3.html#E1:50_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="fib_num3.html#T34" target="_self">Th34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,0</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fib_num3.html#K1" target="_self">Lucas</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="fib_num3.html#K1" target="_self">Lucas</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fib_num3.html#E2:50_2"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,0</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</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">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fib_num3.html#K1" target="_self">Lucas</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="fib_num3.html#K1" target="_self">Lucas</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fib_num2.html#T26">FIB_NUM2:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,0</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</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">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span></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">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fib_num3.html#T12" target="_self">Th12</a></i>
;<br/>
<b>hence </b><a NAME="E4:50_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 2]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:50"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="fib_num.html#S1">FIB_NUM:sch 1</a>(<a class="txt" href="fib_num3.html#E1:50"><i><font color="Green">E32</font></i></a>, <a class="txt" href="fib_num3.html#E2:50"><i><font color="Green">E33</font></i></a>, <a class="txt" href="fib_num3.html#E3:50"><i><font color="Green">E34</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:50"/>
 <a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,0</span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <font color="Maroon">c<sub>2</sub></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> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 ;<br/>


</div>
