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<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">b</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> <a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">b</font>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<a NAME="E1:51"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

 <a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">b</font>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b</font></span>)</span>,0 = 
<font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
 <a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E2:51_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:51"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>proof </b></a><div class="add">

 <a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">b</font>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b</font></span>)</span>,1 = 
<font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,1</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,0</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 2</span>)</span>
<b>by </b><i><a class="txt" href="fib_num3.html#E1:51"><i><font color="Green">E27</font></i></a></i>
<br/>.= 
 <a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>

;<br/>
<b>hence </b><a NAME="E2:51_2"/>
<font color="Maroon">S<sub>1</sub></font>[1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:51"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 2]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:51_3"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 <b>and </b><br/><a NAME="E2:51_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>

 <a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">b</font>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fib_num3.html#T6" target="_self">Th6</a>, <a class="ref" href="fib_num3.html#T7" target="_self">Th7</a>, <a class="txt" href="fib_num3.html#E1:51"><i><font color="Green">E27</font></i></a></i>
<br/>.= 
 <a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 3</span>)</span>
<b>by </b><i><a class="txt" href="fib_num3.html#E2:51"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
 <a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>

;<br/>
<b>hence </b><a NAME="E4:51_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:51"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="fib_num.html#S1">FIB_NUM:sch 1</a>(<a class="ref" href="fib_num3.html#T4" target="_self">Th4</a>, <a class="ref" href="fib_num3.html#T5" target="_self">Th5</a>, <a class="txt" href="fib_num3.html#E7"><i><font color="Green">Lemma34</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:51"/>
 <a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">b</font>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="fib_num3.html#K2" target="_self">GenFib</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>


</div>
