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

<b>let </b><font color="Maroon">F</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:27"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <font color="Olive">n</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">f</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E3:27"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <font color="Olive">n</font>
 <b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<b>set </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1;<br/>
<a NAME="E4:27"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>
<br/>.= 
 <a href="pre_ff.html#K3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="fib_num.html#E1:27"><i><font color="Green">E27</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E2:27_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:27"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>proof </b></a><div class="add">

<a NAME="E1:27_2"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="fib_num.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="txt" href="fib_num.html#E3:27"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</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="txt" href="fib_num.html#E1:27"><i><font color="Green">E27</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E2:27_2_1"/>
<font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_2"/>
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default"><a href="fib_num.html#K1" target="_self">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span>;<br/>
<a NAME="E6:27"/>
 ex <font color="Olive">g</font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>)
 
<b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<b>then consider </b><font color="Maroon">g</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E8:27"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>)
 ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</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">a<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span>;<br/>
<a NAME="E9:27"/>
 ex <font color="Olive">h</font> being  <a href="seq_1.html#NM1">Real_Sequence</a> st <br/>for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">n</font>)
 
<b>from </b><i><a class="ref" href="seq_1.html#S1">SEQ_1:sch 1</a>();<br/></i>
<b>then consider </b><font color="Maroon">h</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E11:27"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">n</font>)
 ;<br/>
<a NAME="E12:27"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">h</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3"><b>proof </b></a><div class="add">

<a NAME="E1:27_3"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
 <a href="pre_ff.html#K3">Fib</a> <font color="Maroon">n</font>
<b>by </b><i><a class="txt" href="fib_num.html#E1:27"><i><font color="Green">E27</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K1" target="_self">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="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">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</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" target="_self">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</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="xcmplx_0.html#K6">-</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">n</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>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T121">XCMPLX_1:121</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</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">n</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>
<b>by </b><i><a class="ref" href="fib_num.html#T2" target="_self">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E2:27_3_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:27_3"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:27_3_2"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="fib_num.html#E8:27"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:27_3_2"/>
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:27_3"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K9">+</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:27"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">h</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">f</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4"><b>proof </b></a><div class="add">

<a NAME="E1:27_4"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">h</font> is <a href="seq_2.html#V3">bounded</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:27_4_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:27_4_1_1"/>
<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</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">n</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:27_4_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_4_1"/>
<font color="Maroon">h</font> is <a href="seq_2.html#V3">bounded</a>
 <b>by </b><i><a class="ref" href="seq_2.html#T15">SEQ_2:15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:27_4"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">f</font> <a href="seq_1.html#K18">"</a>  is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4_2"><b>proof </b></a><div class="add">

<a NAME="E1:27_4_2"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">x</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">x</font> holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:27_4_2_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font>
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:27_4_2_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E4:27_4_2_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E5:27_4_2_1"/><i><font color="Green">E44</font></i>: 
1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>
<a NAME="E6:27_4_2_1"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:27_4_2_1_1"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:27_4_2_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="pre_ff.html#K3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pre_ff.html#K3">Fib</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:27_4_2_1_1"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<a NAME="E4:27_4_2_1_1"/><b>then </b>
 <a href="pre_ff.html#K3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E5:27_4_2_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pre_ff.html#K3">Fib</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="fib_num.html#E1:27_4"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:27_4_2_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="fib_num.html#E1:27"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E7:27_4_2_1_1"/>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T21">NAT_1:21</a></i>;<br/>
<a NAME="E8:27_4_2_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<a NAME="E9:27_4_2_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="fib_num.html#E2:27_4"><i><font color="Green">E40</font></i></a>, <a class="ref" href="real_2.html#T152">REAL_2:152</a></i>;<br/>
<a NAME="E10:27_4_2_1_1"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E11:27_4_2_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="real_2.html#T151">REAL_2:151</a></i>;<br/>
<a NAME="E12:27_4_2_1_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="fib_num.html#E2:27_4"><i><font color="Green">E40</font></i></a>, <a class="txt" href="fib_num.html#E1:27_4_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:27_4_2_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K5">"</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/>
<i><font color="Green">E51</font></i>:  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0</span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D8">SEQ_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K5">"</a> 
<b>by </b><i><a class="txt" href="fib_num.html#E5:27_4_2_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>
;<br/>
<a NAME="E15:27_4_2_1_1"/>
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="txt" href="fib_num.html#E3:27_4_2_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E16:27_4_2_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="fib_num.html#E13:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="fib_num.html#E2:27_4_2_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:27_4_2_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_4_2"/>
<font color="Maroon">f</font> <a href="seq_1.html#K18">"</a>  is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a></i>;<br/>

<b>hence </b><a NAME="E3:27_4_2"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:27_4"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">h</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K18">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D9">SEQ_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E4:27_4"/>
<font color="Maroon">h</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">f</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="fib_num.html#E11:27"><i><font color="Green">E34</font></i></a>, <a class="txt" href="fib_num.html#E12:27"><i><font color="Green">E35</font></i></a>, <a class="ref" href="seq_2.html#T39">SEQ_2:39</a></i>;<br/>

<b>thus </b><a NAME="E5:27_4"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="fib_num.html#E11:27"><i><font color="Green">E34</font></i></a>, <a class="txt" href="fib_num.html#E12:27"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="seq_2.html#T40">SEQ_2:40</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">f2</font> = <font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2;<br/>
<a NAME="E14:27"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">f</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_5"><b>proof </b></a><div class="add">

<a NAME="E1:27_5"/><i><font color="Green">E102</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<i><font color="Green">E103</font></i>: <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>
<br/>.= 
 <a href="pre_ff.html#K3">Fib</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="fib_num.html#E1:27"><i><font color="Green">E27</font></i></a></i>
;<br/>
<a NAME="E2:27_5_1"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T21">NAT_1:21</a></i>;<br/>
<a NAME="E3:27_5_1"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<b>hence </b><a NAME="E4:27_5_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:27_5"/><i><font color="Green">E104</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:27_5_2"/><i><font color="Green">E105</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="fib_num.html#E12:27"><i><font color="Green">E35</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></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"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="fib_num.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>
<br/>.= 
1
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T107">XCMPLX_1:107</a></i>
;<br/>
<b>hence </b><a NAME="E3:27_5_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:27_5"/><b>then </b><i><font color="Green">E106</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> is <a href="funct_1.html#V3">constant</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D6">SEQM_3:def 6</a></i>;<br/>
<a NAME="E4:27_5"/><b>then </b><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T39">SEQ_4:39</a></i>;<br/>
<a NAME="E5:27_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E6:27_5"/><b>then </b><i><font color="Green">E108</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="seq_4.html#T40">SEQ_4:40</a></i>;<br/>
<a NAME="E7:27_5"/><i><font color="Green">E109</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">f</font></span>)</span> <a href="seqm_3.html#K1">^\</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="seqm_3.html#T43">SEQM_3:43</a></i>;<br/>
<b>hence </b><a NAME="E8:27_5"/>
<font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">f</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="fib_num.html#E13:27"><i><font color="Green">E38</font></i></a>, <a class="ref" href="seq_4.html#T35">SEQ_4:35</a></i>;<br/>

<b>thus </b><a NAME="E9:27_5"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="fib_num.html#E13:27"><i><font color="Green">E38</font></i></a>, , , <a class="ref" href="seq_4.html#T36">SEQ_4:36</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:27"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">g</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="fib_num.html#E5:27"><i><font color="Green">E32</font></i></a>, <a class="ref" href="seq_1.html#T57">SEQ_1:57</a></i>;<br/>
<a NAME="E16:27"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">g</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">f</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="txt" href="fib_num.html#E8:27"><i><font color="Green">E33</font></i></a>, <a class="txt" href="fib_num.html#E11:27"><i><font color="Green">E34</font></i></a>, <a class="ref" href="seq_2.html#T19">SEQ_2:19</a></i>;<br/>
<i><font color="Green">E112</font></i>:  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">f</font></span>)</span> = 
1 <a href="nat_1.html#K1">+</a> 0
<b>by </b><i><a class="txt" href="fib_num.html#E8:27"><i><font color="Green">E33</font></i></a>, <a class="txt" href="fib_num.html#E11:27"><i><font color="Green">E34</font></i></a>, <a class="txt" href="fib_num.html#E12:27"><i><font color="Green">E35</font></i></a>, <a class="ref" href="seq_2.html#T20">SEQ_2:20</a></i>
<br/>.= 
1

;<br/>
<b>set </b><font color="Maroon">g1</font> = <font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1;<br/>
<a NAME="E18:27"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">g</font> is <a href="relat_1.html#V2">being_not_0</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_7"><b>proof </b></a><div class="add">

<a NAME="E1:27_7"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<i><font color="Green">E115</font></i>: <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" target="_self">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span>
<b>by </b><i><a class="ref" href="fib_num.html#T2" target="_self">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" target="_self">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"><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>
<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="fib_num.html#K1" target="_self">tau</a>  <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</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>
<b>by </b><i>, <a class="ref" href="power.html#T46">POWER:46</a></i>
;<br/>
<a NAME="E2:27_7_1"/><i><font color="Green">E116</font></i>: 
<a href="fib_num.html#K1" target="_self">tau</a>  <a href="newton.html#K2">|^</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="prepower.html#T12">PREPOWER:12</a></i>;<br/>
<a NAME="E3:27_7_1"/>
<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="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T203">XCMPLX_1:203</a></i>;<br/>
<b>hence </b><a NAME="E4:27_7_1"/>
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, , <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_7"/>
<font color="Maroon">g</font> is <a href="relat_1.html#V2">being_not_0</a>
 <b>by </b><i><a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:27"/><i><font color="Green">E117</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <font color="Maroon">g</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fib_num.html#K1" target="_self">tau</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_8"><b>proof </b></a><div class="add">

<a NAME="E1:27_8"/><i><font color="Green">E118</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="fib_num.html#K1" target="_self">tau</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<i><font color="Green">E119</font></i>: <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" target="_self">tau</a>  <a href="power.html#K3">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span>
<b>by </b><i><a class="ref" href="fib_num.html#T2" target="_self">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" target="_self">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"><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>
<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="fib_num.html#K1" target="_self">tau</a>  <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</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>
<b>by </b><i>, <a class="ref" href="power.html#T46">POWER:46</a></i>
;<br/>
<i><font color="Green">E120</font></i>: <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" target="_self">tau</a>  <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> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> 5</span>)</span>
<b>by </b><i><a class="ref" href="fib_num.html#T2" target="_self">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" target="_self">tau</a>  <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> <a href="xcmplx_0.html#K3">*</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>
<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="fib_num.html#K1" target="_self">tau</a>  <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</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="square_1.html#K7">sqrt</a> 5</span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="power.html#T46">POWER:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" target="_self">tau</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="fib_num.html#K1" target="_self">tau</a>  <a href="newton.html#K2">|^</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="square_1.html#K7">sqrt</a> 5</span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
<a href="fib_num.html#K1" target="_self">tau</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E3:27_8_1"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="fib_num.html#E13:27"><i><font color="Green">E38</font></i></a>, <a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" target="_self">tau</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</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"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="fib_num.html#T1" target="_self">Th1</a></i>
<br/>.= 
<a href="fib_num.html#K1" target="_self">tau</a>  <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span></span>)</span>

<br/>.= 
<a href="fib_num.html#K1" target="_self">tau</a>  <a href="xcmplx_0.html#K3">*</a> 1
<b>by </b><i>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
<br/>.= 
 <a href="fib_num.html#K1" target="_self">tau</a> 

;<br/>
<b>hence </b><a NAME="E5:27_8_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="fib_num.html#K1" target="_self">tau</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:27_8"/><b>then </b><i><font color="Green">E122</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <font color="Maroon">g</font> is <a href="funct_1.html#V3">constant</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D6">SEQM_3:def 6</a></i>;<br/>
<b>hence </b><a NAME="E3:27_8"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <font color="Maroon">g</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_4.html#T39">SEQ_4:39</a></i>;<br/>

<a NAME="E4:27_8"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="fib_num.html#K1" target="_self">tau</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:27_8"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fib_num.html#K1" target="_self">tau</a> 
 <b>by </b><i>, <a class="ref" href="seq_4.html#T40">SEQ_4:40</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:27"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1 is <a href="relat_1.html#V2">being_not_0</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_9"><b>proof </b></a><div class="add">

<a NAME="E1:27_9"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>
<br/>.= 
 <a href="pre_ff.html#K3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="fib_num.html#E1:27"><i><font color="Green">E27</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E2:27_9_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_9"/>
<font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1 is <a href="relat_1.html#V2">being_not_0</a>
 <b>by </b><i><a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:27"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_10"><b>proof </b></a><div class="add">

<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1 = 
<font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#T36">SEQM_3:36</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2

;<br/>
<b>hence </b><a NAME="E2:27_10"/>
<font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">g2</font> = <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1;<br/>
<a NAME="E22:27"/><i><font color="Green">E126</font></i>: 
( <font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1 is <a href="relat_1.html#V2">being_not_0</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1 is <a href="relat_1.html#V2">being_not_0</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_11"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:27_11"/>
<font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1 is <a href="relat_1.html#V2">being_not_0</a>
 <b>by </b><i><a class="txt" href="fib_num.html#E13:27"><i><font color="Green">E38</font></i></a>, <a class="ref" href="seqm_3.html#T40">SEQM_3:40</a></i>;<br/>

<b>hence </b><a NAME="E2:27_11"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1 is <a href="relat_1.html#V2">being_not_0</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#T40">SEQM_3:40</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:27"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2 is <a href="relat_1.html#V2">being_not_0</a>
 
<b>by </b><i>, , <a class="ref" href="seqm_3.html#T40">SEQM_3:40</a></i>;<br/>
<a NAME="E24:27"/><i><font color="Green">E128</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fib_num.html#K1" target="_self">tau</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_12"><b>proof </b></a><div class="add">

<a NAME="E1:27_12"/><i><font color="Green">E129</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:27_12"/><i><font color="Green">E130</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_12_1"><b>proof </b></a><div class="add">

<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1 = 
<font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#T36">SEQM_3:36</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 2

;<br/>
<b>hence </b><a NAME="E2:27_12_1"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 2
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:27_12"/><i><font color="Green">E131</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_12_2"><b>proof </b></a><div class="add">

<a NAME="E1:27_12_2"/><i><font color="Green">E132</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">f</font></span>)</span> <a href="seqm_3.html#K1">^\</a> 2
 
<b>by </b><i>, <a class="ref" href="seqm_3.html#T43">SEQM_3:43</a></i>;<br/>
<a NAME="E2:27_12_2"/><b>then </b><i><font color="Green">E133</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, <a class="ref" href="seq_4.html#T33">SEQ_4:33</a></i>;<br/>
<a NAME="E3:27_12_2"/><i><font color="Green">E134</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, , , <a class="ref" href="seq_4.html#T33">SEQ_4:33</a></i>;<br/>
<a NAME="E4:27_12_2"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_12_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:27_12_2_1"/><i><font color="Green">E135</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/>
<a NAME="E2:27_12_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/>
<a NAME="E3:27_12_2_1"/><b>then </b><i><font color="Green">E136</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K5">"</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T203">XCMPLX_1:203</a></i>;<br/>
<a NAME="E4:27_12_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></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"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="fib_num.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E5:27_12_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, , <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:27_12_2"/><b>then </b><i><font color="Green">E137</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> is <a href="relat_1.html#V2">being_not_0</a>
 
<b>by </b><i><a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/>
<a NAME="E6:27_12_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K18">"</a>  is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i>, , <a class="ref" href="seq_2.html#T35">SEQ_2:35</a></i>;<br/>
<b>hence </b><a NAME="E7:27_12_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_1.html#T48">SEQ_1:48</a></i>;<br/>

 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span></span>)</span> <a href="seq_1.html#K18">"</a> </span>)</span> = 
1 <a href="xcmplx_0.html#K5">"</a> 
<b>by </b><i>, , , <a class="ref" href="seq_2.html#T36">SEQ_2:36</a></i>
<br/>.= 
1

;<br/>
<b>hence </b><a NAME="E9:27_12_2"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="seq_1.html#T48">SEQ_1:48</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:27_12"/><i><font color="Green">E138</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fib_num.html#K1" target="_self">tau</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_12_3"><b>proof </b></a><div class="add">

<a NAME="E1:27_12_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <font color="Maroon">g</font></span>)</span> <a href="seqm_3.html#K1">^\</a> 1
 
<b>by </b><i><a class="ref" href="seqm_3.html#T43">SEQM_3:43</a></i>;<br/>
<b>hence </b><a NAME="E2:27_12_3"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fib_num.html#K1" target="_self">tau</a>  )
 <b>by </b><i>, <a class="ref" href="seq_4.html#T33">SEQ_4:33</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:27_12"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i>, , <a class="ref" href="seq_2.html#T28">SEQ_2:28</a></i>;<br/>

 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> = 
1 <a href="xcmplx_0.html#K3">*</a> <a href="fib_num.html#K1" target="_self">tau</a> 
<b>by </b><i>, , , <a class="ref" href="seq_2.html#T29">SEQ_2:29</a></i>
<br/>.= 
 <a href="fib_num.html#K1" target="_self">tau</a> 

;<br/>
<b>hence </b><a NAME="E7:27_12"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fib_num.html#K1" target="_self">tau</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E25:27"/><i><font color="Green">E139</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_13"><b>proof </b></a><div class="add">

<a NAME="E1:27_13"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K19">/"</a> <font color="Maroon">f</font></span>)</span> <a href="seqm_3.html#K1">^\</a> 1
 
<b>by </b><i><a class="ref" href="seqm_3.html#T43">SEQM_3:43</a></i>;<br/>
<b>hence </b><a NAME="E2:27_13"/>
( <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, , <a class="ref" href="seq_4.html#T33">SEQ_4:33</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E26:27"/><i><font color="Green">E140</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fib_num.html#K1" target="_self">tau</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_14"><b>proof </b></a><div class="add">

<a NAME="E1:27_14"/><i><font color="Green">E141</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E2:27_14"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i>, , <a class="ref" href="seq_2.html#T28">SEQ_2:28</a></i>;<br/>

<a NAME="E3:27_14"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <a href="fib_num.html#K1" target="_self">tau</a>  <a href="xcmplx_0.html#K3">*</a> 1
 
<b>by </b><i>, , , <a class="ref" href="seq_2.html#T29">SEQ_2:29</a></i>;<br/>
<b>hence </b><a NAME="E4:27_14"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fib_num.html#K1" target="_self">tau</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E27:27"/><i><font color="Green">E142</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 2</span>)</span> <a href="seq_1.html#K19">/"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seqm_3.html#K1">^\</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="seqm_3.html#K1">^\</a> 1
 
<b>by </b><i><a class="txt" href="fib_num.html#E4:27"><i><font color="Green">E29</font></i></a>, , <a class="ref" href="seqm_3.html#T43">SEQM_3:43</a></i>;<br/>
<b>hence </b><a NAME="E28:27"/>
<font color="Maroon">F</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i>, <a class="ref" href="seq_4.html#T35">SEQ_4:35</a></i>;<br/>

<b>thus </b><a NAME="E29:27"/>
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="fib_num.html#K1" target="_self">tau</a> 
 <b>by </b><i>, , <a class="ref" href="seq_4.html#T36">SEQ_4:36</a></i>;<br/>


</div>
