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<b>let </b><font color="Maroon" title="c1">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>set </b><font color="Maroon" title="c2">a</font> = <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">n</font>;<br/>
<b>set </b><font color="Maroon" title="c3">b</font> = <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">n</font>;<br/>
<i><font color="Green" title="E24">A1</font></i>: <span class="p1">(<span class="default"><a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 2 = 
<span class="p1">(<span class="default"><a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Maroon" title="c1">n</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> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="ref" href="wsierp_1.html#T2" title="WSIERP_1:th.2">WSIERP_1:2</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">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<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">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="ref" href="fib_num.html#T7" title="FIB_NUM:th.7">FIB_NUM:7</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">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<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">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<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">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fib_num.html#T7" title="FIB_NUM:th.7">FIB_NUM:7</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">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<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">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<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">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T77" title="XCMPLX_1:th.77">XCMPLX_1:77</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<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">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<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">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<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">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<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">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<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">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 5
<b>by </b><i><a class="ref" href="fib_num3.html#T2" target="_self" title="FIB_NUM3:th.2">Th2</a></i>
;<br/>
<i><font color="Green" title="E25">A2</font></i>: <span class="p1">(<span class="default"><a href="fib_num3.html#K1" title="FIB_NUM3:func.1">Lucas</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 2 = 
<span class="p1">(<span class="default"><a href="fib_num3.html#K1" title="FIB_NUM3:func.1">Lucas</a> <font color="Maroon" title="c1">n</font></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#K1" title="FIB_NUM3:func.1">Lucas</a> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="ref" href="wsierp_1.html#T2" title="WSIERP_1:th.2">WSIERP_1:2</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> <font color="Maroon" title="c1">n</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">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="fib_num3.html#K1" title="FIB_NUM3:func.1">Lucas</a> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="ref" href="fib_num3.html#T21" target="_self" title="FIB_NUM3:th.21">Th21</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> <font color="Maroon" title="c1">n</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">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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">n</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">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fib_num3.html#T21" target="_self" title="FIB_NUM3:th.21">Th21</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">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<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">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<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">n</font></span>)</span></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">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">n</font></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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="Maroon" title="c1">n</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a> </span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="fib_num3.html#T8" target="_self" title="FIB_NUM3:th.8">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4</span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 5</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 4</span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="fib_num3.html#T2" target="_self" title="FIB_NUM3:th.2">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">n</font>

;<br/>
<b>then </b><span class="p1">(<span class="default">5 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num3.html#K1" title="FIB_NUM3:func.1">Lucas</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 2</span>)</span> = 
<span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="txt" href="fib_num3.html#E1:34"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="fib_num3.html#E2:34"><i><font color="Green" title="E25">A2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>
<br/>.= 
4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>

<br/>.= 
4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</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#E4"><i><font color="Green" title="E4">Lm1</font></i></a>, <a class="ref" href="fib_num2.html#T7" title="FIB_NUM2:th.7">FIB_NUM2:7</a></i>
;<br/>
<b>hence </b><a NAME="E5:34"/>
<span class="p1">(<span class="default">5 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num3.html#K1" title="FIB_NUM3:func.1">Lucas</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 ;<br/>


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