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

<b>set </b><font color="Maroon">a</font> = <a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">b</font> = <a href="fib_num.html#K2">tau_bar</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font>;<br/>
<i><font color="Green">E27</font></i>: <span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <font color="Maroon">n</font></span>)</span> <a href="prepower.html#K3">|^</a> 2 = 
<span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="wsierp_1.html#T2">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">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="fib_num.html#K2">tau_bar</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="fib_num.html#T7">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">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="fib_num.html#K2">tau_bar</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</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">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="fib_num.html#K2">tau_bar</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fib_num.html#T7">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">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="fib_num.html#K2">tau_bar</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="fib_num.html#K2">tau_bar</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T77">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">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><a href="fib_num.html#K2">tau_bar</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fib_num.html#K2">tau_bar</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="fib_num.html#K2">tau_bar</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 5
<b>by </b><i/>
;<br/>
<i><font color="Green">E28</font></i>: <span class="p1">(<span class="default"><a href="fib_num3.html#K1" target="_self">Lucas</a> <font color="Maroon">n</font></span>)</span> <a href="prepower.html#K3">|^</a> 2 = 
<span class="p1">(<span class="default"><a href="fib_num3.html#K1" target="_self">Lucas</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="fib_num3.html#K1" target="_self">Lucas</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="wsierp_1.html#T2">WSIERP_1:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="fib_num.html#K2">tau_bar</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="fib_num3.html#K1" target="_self">Lucas</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="fib_num.html#K2">tau_bar</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="fib_num.html#K2">tau_bar</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><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">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="fib_num.html#K2">tau_bar</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K2">tau_bar</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="fib_num.html#K2">tau_bar</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span>

;<br/>
<i><font color="Green">E40</font></i>: <span class="p1">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="fib_num.html#K2">tau_bar</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="fib_num.html#K1">tau</a>  <a href="xcmplx_0.html#K3">*</a> <a href="fib_num.html#K2">tau_bar</a> </span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">n</font>
<b>by </b><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">*</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 4</span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="fib_num.html#D1">FIB_NUM:def 1</a>, <a class="ref" href="fib_num.html#D2">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">-</a> 5</span>)</span> <a href="xcmplx_0.html#K7">/</a> 4</span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">n</font>

;<br/>
<span class="p1">(<span class="default">5 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <font color="Maroon">n</font></span>)</span> <a href="prepower.html#K3">|^</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num3.html#K1" target="_self">Lucas</a> <font color="Maroon">n</font></span>)</span> <a href="prepower.html#K3">|^</a> 2</span>)</span> = 
<span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="fib_num3.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="power.html#K3">to_power</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
4 <a href="xcmplx_0.html#K3">*</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">-</a> 1</span>)</span> <a href="power.html#K3">to_power</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span></span>)</span>

<br/>.= 
4 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="fib_num2.html#T7">FIB_NUM2:7</a></i>
;<br/>
<b>hence </b><a NAME="E5:34"/>
<span class="p1">(<span class="default">5 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <font color="Maroon">n</font></span>)</span> <a href="prepower.html#K3">|^</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num3.html#K1" target="_self">Lucas</a> <font color="Maroon">n</font></span>)</span> <a href="prepower.html#K3">|^</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 4 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/>


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