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<div class="add">

<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b> <a href="fib_num3.html#K2" title="FIB_NUM3:func.2">GenFib</a> <font color="Maroon" title="c2">b</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span>,$1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fib_num3.html#K2" title="FIB_NUM3:func.2">GenFib</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p1">(<span class="default">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/>
<a NAME="E1:51"/><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<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" title="FIB_NUM3:func.2">GenFib</a> <font color="Maroon" title="c2">b</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<font color="Maroon" title="c2">b</font>
<b>by </b><i><a class="ref" href="fib_num3.html#T32" target="_self" title="FIB_NUM3:th.32">Th32</a></i>
<br/>.= 
 <a href="fib_num3.html#K2" title="FIB_NUM3:func.2">GenFib</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="fib_num3.html#T32" target="_self" title="FIB_NUM3:th.32">Th32</a></i>
;<br/>
<b>hence </b><a NAME="E2:51_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:51"/><i><font color="Green" title="E33">A2</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" title="FIB_NUM3:func.2">GenFib</a> <font color="Maroon" title="c2">b</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span>,1 = 
<font color="Maroon" title="c1">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">b</font>
<b>by </b><i><a class="ref" href="fib_num3.html#T32" target="_self" title="FIB_NUM3:th.32">Th32</a></i>
<br/>.= 
<font color="Maroon" title="c1">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="fib_num3.html#K2" title="FIB_NUM3:func.2">GenFib</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="fib_num3.html#T32" target="_self" title="FIB_NUM3:th.32">Th32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num3.html#K2" title="FIB_NUM3:func.2">GenFib</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="fib_num3.html#K2" title="FIB_NUM3:func.2">GenFib</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fib_num3.html#T32" target="_self" title="FIB_NUM3:th.32">Th32</a></i>
<br/>.= 
 <a href="fib_num3.html#K2" title="FIB_NUM3:func.2">GenFib</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="fib_num3.html#T34" target="_self" title="FIB_NUM3:th.34">Th34</a></i>
<br/>.= 
 <a href="fib_num3.html#K2" title="FIB_NUM3:func.2">GenFib</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</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" title="E34">A3</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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" title="c4">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:51_3"/><i><font color="Green" title="E35">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>]
 <b>and </b><br/><a NAME="E2:51_3"/><i><font color="Green" title="E36">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ;<br/>

 <a href="fib_num3.html#K2" title="FIB_NUM3:func.2">GenFib</a> <font color="Maroon" title="c2">b</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><a href="fib_num3.html#K2" title="FIB_NUM3:func.2">GenFib</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="fib_num3.html#K2" title="FIB_NUM3:func.2">GenFib</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fib_num3.html#E1:51_3"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="fib_num3.html#E2:51_3"><i><font color="Green" title="E36">A5</font></i></a>, <a class="ref" href="fib_num3.html#T34" target="_self" title="FIB_NUM3:th.34">Th34</a></i>
<br/>.= 
 <a href="fib_num3.html#K2" title="FIB_NUM3:func.2">GenFib</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span>
<b>by </b><i><a class="ref" href="fib_num3.html#T35" target="_self" title="FIB_NUM3:th.35">Th35</a></i>
<br/>.= 
 <a href="fib_num3.html#K2" title="FIB_NUM3:func.2">GenFib</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

;<br/>
<b>hence </b><a NAME="E4:51_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:51"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="fib_num.html#S1" title="FIB_NUM:sch.1">FIB_NUM:sch 1</a>(<a class="txt" href="fib_num3.html#E1:51"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="fib_num3.html#E2:51"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="fib_num3.html#E3:51"><i><font color="Green" title="E34">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:51"/>
 <a href="fib_num3.html#K2" title="FIB_NUM3:func.2">GenFib</a> <font color="Maroon" title="c2">b</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span>,<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fib_num3.html#K2" title="FIB_NUM3:func.2">GenFib</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/>


</div>
