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

<b>let </b><font color="Maroon">c<sub>1</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_num2.html#K5" target="_self">EvenFibs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fib_num2.html#K5" target="_self">EvenFibs</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><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">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span>;<br/>
<a NAME="E1:76"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="fib_num2.html#T23" target="_self">Th23</a>, <a class="ref" href="fib_num2.html#T55" target="_self">Th55</a>, <a class="ref" href="fib_num2.html#T57" target="_self">Th57</a>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/>
<a NAME="E2:76"/><i><font color="Green">E57</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>] 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> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:76_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="fib_num2.html#K5" target="_self">EvenFibs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><a href="fib_num2.html#K5" target="_self">EvenFibs</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><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">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <a href="fib_num2.html#K2" target="_self">FIB</a>  <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="fib_num2.html#K3" target="_self">EvenNAT</a>  <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span> as   <a href="finseq_1.html#NM4">FinSubsequence</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default">1,<span class="p3">(<span class="default"><a href="fib_num2.html#K2" target="_self">FIB</a>  <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="tarski.html#K1">}</a></span> as   <a href="finseq_1.html#NM4">FinSubsequence</a> <b>by </b><i><a class="ref" href="fib_num2.html#T19" target="_self">Th19</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span> <a href="pnproc_1.html#K14">Shift</a> <font color="Maroon">c<sub>6</sub></font>;<br/>
<a NAME="E4:76_1"/><i><font color="Green">E58</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <a href="fib_num2.html#K3" target="_self">EvenNAT</a>  <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T87">RELAT_1:87</a></i>;<br/>
<a NAME="E5:76_1"/>
<a href="fib_num2.html#K3" target="_self">EvenNAT</a>  <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E6:76_1"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="fib_num2.html#E4:76_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E7:76_1"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E9:76_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span> )
 <b>by </b><i><a class="txt" href="fib_num2.html#E6:76_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="fib_num2.html#T21" target="_self">Th21</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E11:76_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="fib_num2.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E12:76_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3
 
<b>by </b><i><a class="txt" href="fib_num2.html#E9:76_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="finseq_1.html#NM4">FinSubsequence</a><b> such that </b><br/><a NAME="E14:76_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span> <a href="pnproc_1.html#K14">Shift</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E15:76_1"/><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="fib_num2.html#E9:76_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="fib_num2.html#E11:76_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="pnproc_1.html#T82">PNPROC_1:82</a></i>;<br/>
<a NAME="E16:76_1"/>
1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4
 
;<br/>
<a NAME="E17:76_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span> <a href="pnproc_1.html#K14">Shift</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="fib_num2.html#K2" target="_self">FIB</a>  <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="fib_num2.html#T20" target="_self">Th20</a></i>;<br/>
<span class="p1">(<span class="default"><a href="fib_num2.html#K5" target="_self">EvenFibs</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><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">2 <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span> = 
<span class="p1">(<span class="default"><a href="fib_num2.html#K1" target="_self">Prefix</a> <a href="fib_num2.html#K2" target="_self">FIB</a> ,<span class="p2">(<span class="default"><a href="fib_num2.html#K3" target="_self">EvenNAT</a>  <a href="xboole_0.html#K3">/\</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><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">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <span class="p2">(<span class="default"><a href="fib_num2.html#K2" target="_self">FIB</a>  <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><a href="fib_num2.html#K3" target="_self">EvenNAT</a>  <a href="xboole_0.html#K3">/\</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><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">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <span class="p2">(<span class="default"><a href="fib_num2.html#K2" target="_self">FIB</a>  <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><a href="fib_num2.html#K3" target="_self">EvenNAT</a>  <a href="xboole_0.html#K3">/\</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="fib_num2.html#K2" target="_self">FIB</a>  <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 4</span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span>
<b>by </b><i><a class="ref" href="fib_num2.html#D2" target="_self">Def2</a></i>
<br/>.= 
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="fib_num2.html#E15:76_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="pnproc_1.html#T3">PNPROC_1:3</a></i>
<br/>.= 
 <a href="fib_num2.html#K5" target="_self">EvenFibs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>
<b>by </b><i><a class="txt" href="fib_num2.html#E14:76_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="fib_num2.html#E17:76_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="fib_num2.html#T60" target="_self">Th60</a></i>
;<br/>
<b>hence </b><a NAME="E19:76_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:76"/>
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="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="fib_num2.html#E1:76"><i><font color="Green">E56</font></i></a>, <a class="txt" href="fib_num2.html#E2:76"><i><font color="Green">E57</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:76"/>
 <a href="fib_num2.html#K5" target="_self">EvenFibs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fib_num2.html#K5" target="_self">EvenFibs</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><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">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span>
 ;<br/>


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