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

<b>set </b><font color="Maroon">c<sub>1</sub></font> = <span class="p1"><a href="tarski.html#K2">{</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> 1</span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default">3,<span class="p3">(<span class="default"><a href="fib_num2.html#K2" target="_self">FIB</a>  <a href="funct_2.html#K3">.</a> 3</span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="tarski.html#K2">}</a></span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1"><a href="tarski.html#K2">{</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> 1</span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default">3,<span class="p3">(<span class="default"><a href="fib_num2.html#K2" target="_self">FIB</a>  <a href="funct_2.html#K3">.</a> 3</span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="tarski.html#K2">}</a></span> as   <a href="finseq_1.html#NM4">FinSubsequence</a> <b>by </b><i><a class="ref" href="fib_num2.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E2:78"/>
( 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; 3 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 
;<br/>
<a NAME="E3:78"/><b>then </b><i><font color="Green">E58</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <a href="fib_num2.html#K2" target="_self">FIB</a>  &amp; 3 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <a href="fib_num2.html#K2" target="_self">FIB</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:78"/><i><font color="Green">E59</font></i>: 
<a href="fib_num2.html#K4" target="_self">OddNAT</a>  <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default">1,2,3</span><a href="domain_1.html#K8">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,3</span><a href="domain_1.html#K7">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:78_1"/>
<a href="fib_num2.html#K4" target="_self">OddNAT</a>  <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default">1,2,3</span><a href="domain_1.html#K8">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,3</span><a href="domain_1.html#K7">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:78_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <a href="fib_num2.html#K4" target="_self">OddNAT</a>  <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default">1,2,3</span><a href="domain_1.html#K8">}</a></span>
 ;<br/><a NAME="E2:78_1_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="fib_num2.html#K4" target="_self">OddNAT</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default">1,2,3</span><a href="domain_1.html#K8">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:78_1_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> 1 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="fib_num2.html#E2:78_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:78_1_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:78_1_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,3</span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:78_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:78_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,3</span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="txt" href="fib_num2.html#E2:78_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="fib_num2.html#T54" target="_self">Th54</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:78_1_1_1_1_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:78_1_1_1_1_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,3</span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:78_1_1"/>
<a href="fib_num2.html#K4" target="_self">OddNAT</a>  <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default">1,2,3</span><a href="domain_1.html#K8">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,3</span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:78_1"/>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,3</span><a href="domain_1.html#K7">}</a></span> <a href="tarski.html#R1">c=</a> <a href="fib_num2.html#K4" target="_self">OddNAT</a>  <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default">1,2,3</span><a href="domain_1.html#K8">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:78_1_2_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,3</span><a href="domain_1.html#K7">}</a></span>
 ;<br/><a NAME="E2:78_1_2_1"/><b>then </b><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 3 )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E3:78_1_2_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="fib_num2.html#E1:78_1_2_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E4:78_1_2_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="fib_num2.html#K4" target="_self">OddNAT</a> 
 ;<br/><a NAME="E5:78_1_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default">1,2,3</span><a href="domain_1.html#K8">}</a></span>
 <b>by </b><i><a class="txt" href="fib_num2.html#E2:78_1_2_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><b>hence </b><a NAME="E6:78_1_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <a href="fib_num2.html#K4" target="_self">OddNAT</a>  <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default">1,2,3</span><a href="domain_1.html#K8">}</a></span>
 <b>by </b><i><a class="txt" href="fib_num2.html#E4:78_1_2_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:78_1_2"/>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,3</span><a href="domain_1.html#K7">}</a></span> <a href="tarski.html#R1">c=</a> <a href="fib_num2.html#K4" target="_self">OddNAT</a>  <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default">1,2,3</span><a href="domain_1.html#K8">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<i><font color="Green">E60</font></i>: <a href="fib_num2.html#K2" target="_self">FIB</a>  <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><span class="p2"><a href="seq_4.html#K1">{</a><span class="default">1</span><a href="seq_4.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="seq_4.html#K1">{</a><span class="default">3</span><a href="seq_4.html#K1">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="fib_num2.html#K2" target="_self">FIB</a>  <a href="relat_1.html#K7">|</a> <span class="p2"><a href="seq_4.html#K1">{</a><span class="default">1</span><a href="seq_4.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="fib_num2.html#K2" target="_self">FIB</a>  <a href="relat_1.html#K7">|</a> <span class="p2"><a href="seq_4.html#K1">{</a><span class="default">3</span><a href="seq_4.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T107">RELAT_1:107</a></i>
<br/>.= 
<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> 1</span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="fib_num2.html#K2" target="_self">FIB</a>  <a href="relat_1.html#K7">|</a> <span class="p2"><a href="seq_4.html#K1">{</a><span class="default">3</span><a href="seq_4.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="fib_num2.html#E3:78"><i><font color="Green">E58</font></i></a>, <a class="ref" href="grfunc_1.html#T89">GRFUNC_1:89</a></i>
<br/>.= 
<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> 1</span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</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">3,<span class="p3">(<span class="default"><a href="fib_num2.html#K2" target="_self">FIB</a>  <a href="funct_2.html#K3">.</a> 3</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="txt" href="fib_num2.html#E3:78"><i><font color="Green">E58</font></i></a>, <a class="ref" href="grfunc_1.html#T89">GRFUNC_1:89</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="enumset1.html#T41">ENUMSET1:41</a></i>
;<br/>
 <a href="fib_num2.html#K6" target="_self">OddFibs</a> 3 = 
 <a href="fib_num2.html#K1" target="_self">Prefix</a> <a href="fib_num2.html#K2" target="_self">FIB</a> ,<span class="p1">(<span class="default"><a href="fib_num2.html#K4" target="_self">OddNAT</a>  <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 3</span>)</span></span>)</span>

<br/>.= 
 <a href="finseq_1.html#K15">Seq</a> <span class="p1">(<span class="default"><a href="fib_num2.html#K2" target="_self">FIB</a>  <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="fib_num2.html#K4" target="_self">OddNAT</a>  <a href="xboole_0.html#K3">/\</a> <span class="p3"><a href="domain_1.html#K8">{</a><span class="default">1,2,3</span><a href="domain_1.html#K8">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_3.html#T1">FINSEQ_3:1</a></i>
<br/>.= 
 <a href="finseq_1.html#K15">Seq</a> <span class="p1">(<span class="default"><a href="fib_num2.html#K2" target="_self">FIB</a>  <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><span class="p3"><a href="seq_4.html#K1">{</a><span class="default">1</span><a href="seq_4.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="seq_4.html#K1">{</a><span class="default">3</span><a href="seq_4.html#K1">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fib_num2.html#E4:78"><i><font color="Green">E59</font></i></a>, <a class="ref" href="enumset1.html#T41">ENUMSET1:41</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10">&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> 1</span>)</span>,<span class="p2">(<span class="default"><a href="fib_num2.html#K2" target="_self">FIB</a>  <a href="funct_2.html#K3">.</a> 3</span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
<b>by </b><i><a class="txt" href="fib_num2.html#E5:78"><i><font color="Green">E60</font></i></a>, <a class="ref" href="fib_num2.html#T18" target="_self">Th18</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pre_ff.html#K3">Fib</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="fib_num2.html#K2" target="_self">FIB</a>  <a href="funct_2.html#K3">.</a> 3</span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
<b>by </b><i><a class="ref" href="fib_num2.html#D2" target="_self">Def2</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default">1,2</span><a href="finseq_1.html#K10">*&gt;</a></span>
<b>by </b><i><a class="ref" href="fib_num2.html#D2" target="_self">Def2</a>, <a class="ref" href="fib_num2.html#T24" target="_self">Th24</a>, <a class="ref" href="pre_ff.html#T1">PRE_FF:1</a></i>
;<br/>
<b>hence </b><a NAME="E7:78"/>
 <a href="fib_num2.html#K6" target="_self">OddFibs</a> 3 <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default">1,2</span><a href="finseq_1.html#K10">*&gt;</a></span>
 ;<br/>


</div>
