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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="square_1.html#K7">sqrt</a> 5;<br/>
<a NAME="E1:23"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="fib_num.html#E23"><i><font color="Green">Lemma18</font></i></a>, <a class="ref" href="power.html#T46">POWER:46</a></i>;<br/>
<a NAME="E2:23"/><b>then </b><i><font color="Green">E23</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="txt" href="fib_num.html#E24"><i><font color="Green">Lemma19</font></i></a>, <a class="txt" href="fib_num.html#E26"><i><font color="Green">Lemma21</font></i></a></i>;<br/>
<a NAME="E3:23"/><i><font color="Green">E24</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<i><font color="Green">E25</font></i>:  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font></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="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
;<br/>
<a NAME="E5:23"/>
 <a href="square_1.html#K7">sqrt</a> 5 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="fib_num.html#E18"><i><font color="Green">Lemma14</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:23"/><b>then </b><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span> <a href="xcmplx_0.html#K5">"</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/>
<a NAME="E7:23"/><b>then </b><i><font color="Green">E27</font></i>: 
 <a href="complex1.html#K18">abs</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#K5">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span> <a href="xcmplx_0.html#K5">"</a> 
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E8:23"/>
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K7">/</a> 2
 
<b>by </b><i><a class="txt" href="fib_num.html#E18"><i><font color="Green">Lemma14</font></i></a>, <a class="ref" href="real_2.html#T151">REAL_2:151</a></i>;<br/>
<a NAME="E9:23"/><b>then </b>
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E10:23"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="complex1.html#K18">abs</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#K5">"</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="fib_num.html#E7:23"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>;<br/>
<a NAME="E11:23"/>
 <a href="complex1.html#K18">abs</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#K5">"</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="fib_num.html#E6:23"><i><font color="Green">E26</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<b>hence </b><a NAME="E12:23"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K2" target="_self">tau_bar</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font></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="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <b>by </b><i><a class="txt" href="fib_num.html#E2:23"><i><font color="Green">E23</font></i></a>, <a class="txt" href="fib_num.html#E3:23"><i><font color="Green">E24</font></i></a>, <a class="txt" href="fib_num.html#E4:23"><i><font color="Green">E25</font></i></a>, <a class="txt" href="fib_num.html#E10:23"><i><font color="Green">E28</font></i></a>, <a class="ref" href="real_2.html#T139">REAL_2:139</a></i>;<br/>


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