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<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="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><a href="fib_num2.html#K5" target="_self">EvenFibs</a> <span class="p2">(<span class="default"><span class="p3">(<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> <a href="hidden.html#R1">=</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">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> 3</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1;<br/>
 <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><a href="fib_num2.html#K5" target="_self">EvenFibs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> = 
 <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><a href="fib_num2.html#K5" target="_self">EvenFibs</a> 2</span>)</span>

<br/>.= 
 <a href="wsierp_1.html#K8">Sum</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">1</span><a href="binarith.html#K13">*&gt;</a></span>
<b>by </b><i><a class="ref" href="fib_num2.html#T57" target="_self">Th57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</a> 3</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i><a class="ref" href="fib_num2.html#T24" target="_self">Th24</a>, <a class="ref" href="rvsum_1.html#T103">RVSUM_1:103</a></i>
<br/>.= 
<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">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1

;<br/>
<a NAME="E2:82"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E3:82"/><i><font color="Green">E62</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="82_2"><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>reconsider </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">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T27">FINSEQ_2:27</a></i>;<br/>
<b>assume </b><a NAME="E2:82_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 ;<br/>

 <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><a href="fib_num2.html#K5" target="_self">EvenFibs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<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> = 
 <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num2.html#K5" target="_self">EvenFibs</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<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="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p0">(<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>)</span>
<b>by </b><i><a class="ref" href="fib_num2.html#T61" target="_self">Th61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.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">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<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>
<b>by </b><i><a class="ref" href="rvsum_1.html#T104">RVSUM_1:104</a></i>
<br/>.= 
<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"><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> 3</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.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">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>
<b>by </b><i><a class="txt" href="fib_num2.html#E2:82_2"><i><font color="Green">E63</font></i></a></i>
<br/>.= 
<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"><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> 3</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">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>)</span> <a href="xcmplx_0.html#K6">-</a> 1

<br/>.= 
<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">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> 5</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i><a class="ref" href="fib_num2.html#T29" target="_self">Th29</a></i>
;<br/>
<b>hence </b><a NAME="E4:82_2"/>
<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="E4:82"/>
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#E2:82"><i><font color="Green">E61</font></i></a>, <a class="txt" href="fib_num2.html#E3:82"><i><font color="Green">E62</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:82"/>
 <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><a href="fib_num2.html#K5" target="_self">EvenFibs</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>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</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">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> 3</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/>


</div>
