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<div class="add">

<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:57"/><i><font color="Green" title="E43">A1</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/>

<a NAME="E2:57"/><i><font color="Green" title="E44">A2</font></i>: 
not <font color="Maroon" title="c1">n</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 
<b>by </b><i><a class="txt" href="fib_num2.html#E1:57"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="nat_2.html#D1" title="NAT_2:def.1">NAT_2:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b> <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> $1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <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="E3:57"/><i><font color="Green" title="E45">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[2]
 
<b>by </b><i><a class="ref" href="fib_num2.html#T23" target="_self" title="FIB_NUM2:th.23">Th23</a>, <a class="ref" href="fib_num2.html#T24" target="_self" title="FIB_NUM2:th.24">Th24</a></i>;<br/>
<a NAME="E4:57"/><i><font color="Green" title="E46">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[3]
 
<b>by </b><i><a class="ref" href="fib_num2.html#T24" target="_self" title="FIB_NUM2:th.24">Th24</a>, <a class="ref" href="fib_num2.html#T25" target="_self" title="FIB_NUM2:th.25">Th25</a></i>;<br/>
<a NAME="E5:57"/><i><font color="Green" title="E47">A5</font></i>: 
for <font color="Olive" title="b1">k</font> being non <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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="57_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">k</font> be  non <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:57_1"/><i><font color="Green" title="E48">A6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font>]
 ;<br/>

<b>assume </b><a NAME="E2:57_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ;<br/>

<a NAME="E3:57_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">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="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">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_num2.html#E1:57_1"><i><font color="Green" title="E48">A6</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E4:57_1"/><b>then </b>
 <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">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="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">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="ref" href="fib_num2.html#T26" target="_self" title="FIB_NUM2:th.26">Th26</a></i>;<br/>
<a NAME="E5:57_1"/><b>then </b>
 <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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_num2.html#T27" target="_self" title="FIB_NUM2:th.27">Th27</a></i>;<br/>
<b>hence </b><a NAME="E6:57_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:57"/>
for <font color="Olive" title="b1">n</font> being non <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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">n</font>]
 
<b>from </b><i><a class="ref" href="fib_num2.html#S2" target="_self" title="FIB_NUM2:sch.2">FIB_NUM2:sch 2</a>(<a class="txt" href="fib_num2.html#E3:57"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="fib_num2.html#E4:57"><i><font color="Green" title="E46">A4</font></i></a>, <a class="txt" href="fib_num2.html#E5:57"><i><font color="Green" title="E47">A5</font></i></a>);<br/></i>
<b>hence </b><a NAME="E7:57"/>
 <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="fib_num2.html#E2:57"><i><font color="Green" title="E44">A2</font></i></a></i>;<br/>


</div>
