<?xml version="1.0"?>
<div class="add">

<i><font color="Green">E14</font></i>:  <a href="quin_1.html#K1">delta</a> 1,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="nat_1.html#K2">*</a> 1</span>)</span> <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>
<b>by </b><i><a class="ref" href="quin_1.html#D1">QUIN_1:def 1</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 4</span>)</span>

<br/>.= 
5

;<br/>
<a NAME="E2:16"/><b>then </b><i><font color="Green">E15</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K1">delta</a> 1,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span>,<span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="fib_num.html#K2" target="_self">tau_bar</a> 
 
;<br/>
<a NAME="E3:16"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="fib_num.html#K1" target="_self">tau</a>  or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="fib_num.html#K2" target="_self">tau_bar</a>  ) holds <br/><font color="Olive">b<sub>1</sub></font> <a href="square_1.html#K3">^2</a>  <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:16_2"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="fib_num.html#K1" target="_self">tau</a>  or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="fib_num.html#K2" target="_self">tau_bar</a>  )
 ;<br/>

<a NAME="E2:16_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </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="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="fib_num.html#E1:16"><i><font color="Green">E14</font></i></a>, <a class="txt" href="fib_num.html#E2:16"><i><font color="Green">E15</font></i></a>, <a class="ref" href="fib_num.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E3:16_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:16"/>
<a href="fib_num.html#K1" target="_self">tau</a>  <a href="square_1.html#K3">^2</a>  <a href="hidden.html#R1">=</a> <a href="fib_num.html#K1" target="_self">tau</a>  <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/>

<b>thus </b><a NAME="E5:16"/>
<a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="square_1.html#K3">^2</a>  <a href="hidden.html#R1">=</a> <a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="txt" href="fib_num.html#E3:16"><i><font color="Green">E16</font></i></a></i>;<br/>


</div>
