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

<b>set </b><font color="Maroon">q</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">2,<span class="p3">(<span class="default"><a href="fib_num2.html#K2" target="_self">FIB</a>  <a href="funct_2.html#K3">.</a> 2</span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="tarski.html#K1">}</a></span>;<br/>
<b>reconsider </b><font color="Maroon">q</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">2,<span class="p3">(<span class="default"><a href="fib_num2.html#K2" target="_self">FIB</a>  <a href="funct_2.html#K3">.</a> 2</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/>;<br/>
<a NAME="E2:72"/>
2 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<a NAME="E3:72"/><b>then </b><i><font color="Green">E34</font></i>: 
2 <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="E4:72"/><i><font color="Green">E35</font></i>: 
<a href="fib_num2.html#K3" target="_self">EvenNAT</a>  <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,2</span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">2</span><a href="seq_4.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:72_1"/>
<a href="fib_num2.html#K3" target="_self">EvenNAT</a>  <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,2</span><a href="domain_1.html#K7">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">2</span><a href="seq_4.html#K1">}</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="72_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:72_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="fib_num2.html#K3" target="_self">EvenNAT</a>  <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,2</span><a href="domain_1.html#K7">}</a></span>
 ;<br/><a NAME="E2:72_1_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="fib_num2.html#K3" target="_self">EvenNAT</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,2</span><a href="domain_1.html#K7">}</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="72_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="E1:72_1_1_1_1"/>
( <font color="Maroon">x</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">x</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:72_1_1_1_1_1"/>
<font color="Maroon">x</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="E2:72_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">2</span><a href="seq_4.html#K1">}</a></span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:72_1_1_1_1_2"/>
<font color="Maroon">x</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="E2:72_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">2</span><a href="seq_4.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:72_1_1"/>
<a href="fib_num2.html#K3" target="_self">EvenNAT</a>  <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,2</span><a href="domain_1.html#K7">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">2</span><a href="seq_4.html#K1">}</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="E2:72_1"/>
<span class="p1"><a href="seq_4.html#K1">{</a><span class="default">2</span><a href="seq_4.html#K1">}</a></span> <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"><a href="domain_1.html#K7">{</a><span class="default">1,2</span><a href="domain_1.html#K7">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:72_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">2</span><a href="seq_4.html#K1">}</a></span>
 ;<br/><a NAME="E2:72_1_2_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:72_1_2_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="fib_num2.html#K3" target="_self">EvenNAT</a> 
 ;<br/><a NAME="E4:72_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,2</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/><b>hence </b><a NAME="E5:72_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="fib_num2.html#K3" target="_self">EvenNAT</a>  <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">1,2</span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i>, <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="E2:72_1_2"/>
<span class="p1"><a href="seq_4.html#K1">{</a><span class="default">2</span><a href="seq_4.html#K1">}</a></span> <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"><a href="domain_1.html#K7">{</a><span class="default">1,2</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>


</div><b>end;</b></div>
 <a href="fib_num2.html#K5" target="_self">EvenFibs</a> 2 = 
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, , , <a class="ref" href="grfunc_1.html#T89">GRFUNC_1:89</a></i>
<br/>.= 
<span class="p1"><a href="binarith.html#K11">&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> 2</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
<b>by </b><i><a class="ref" href="pnproc_1.html#T3">PNPROC_1:3</a></i>
<br/>.= 
<span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span>
<b>by </b><i><a class="ref" href="fib_num2.html#T14" target="_self">Th14</a>, </i>
;<br/>
<b>hence </b><a NAME="E6:72"/>
 <a href="fib_num2.html#K5" target="_self">EvenFibs</a> 2 <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11">*&gt;</a></span>
 ;<br/>


</div>
