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<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="fib_num3.html#K1" title="FIB_NUM3:func.1">Lucas</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> $1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> $1</span>)</span>;<br/>
<a NAME="E1:25"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<a NAME="E1:25_1"/>
<a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="power.html#T29" title="POWER:th.29">POWER:29</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> 0</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> 0</span>)</span> = 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="ref" href="power.html#T29" title="POWER:th.29">POWER:29</a></i>
<br/>.= 
2

;<br/>
<b>hence </b><a NAME="E3:25_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i><a class="ref" href="fib_num3.html#T11" target="_self" title="FIB_NUM3:th.11">Th11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:25"/><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>proof </b></a><div class="add">

<a NAME="E1:25_2"/>
( <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a>  &amp; <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  )
 
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>;<br/>
<b>hence </b><a NAME="E2:25_2"/>
<font color="Maroon">S<sub>1</sub></font>[1]
 <b>by </b><i><a class="ref" href="fib_num3.html#T11" target="_self" title="FIB_NUM3:th.11">Th11</a>, <a class="ref" href="fib_num.html#D1" title="FIB_NUM:def.1">FIB_NUM:def 1</a>, <a class="ref" href="fib_num.html#D2" title="FIB_NUM:def.2">FIB_NUM:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:25"/><i><font color="Green" title="E24">A3</font></i>: 
for <font color="Olive" title="b1">k</font> being   <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#K1" title="NAT_1:func.1">+</a> 1] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">k</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><b>that </b><br/><a NAME="E1:25_3"/><i><font color="Green" title="E25">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font>]
 <b>and </b><br/><a NAME="E2:25_3"/><i><font color="Green" title="E26">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ;<br/>

 <a href="fib_num3.html#K1" title="FIB_NUM3:func.1">Lucas</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="fib_num3.html#K1" title="FIB_NUM3:func.1">Lucas</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fib_num3.html#E1:25_3"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="fib_num3.html#T12" target="_self" title="FIB_NUM3:th.12">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fib_num3.html#E2:25_3"><i><font color="Green" title="E26">A5</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fib_num3.html#T9" target="_self" title="FIB_NUM3:th.9">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fib_num3.html#T10" target="_self" title="FIB_NUM3:th.10">Th10</a></i>
;<br/>
<b>hence </b><a NAME="E4:25_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:25"/>
for <font color="Olive" title="b1">k</font> being   <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">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#E1:25"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="fib_num3.html#E2:25"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="fib_num3.html#E3:25"><i><font color="Green" title="E24">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:25"/>
for <font color="Olive" title="b1">n</font> being   <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   <a href="fib_num3.html#K1" title="FIB_NUM3:func.1">Lucas</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Olive" title="b1">n</font></span>)</span>
 ;<br/>


</div>
