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

<b>let </b><font color="Maroon" title="c1">F</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c1">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Olive" title="b1">n</font></span>)</span> ) implies ( <font color="Maroon" title="c1">F</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  ) )</span><br/>

<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E25">A1</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c1">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Olive" title="b1">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">F</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  )</span><br/>

<b>consider </b><font color="Maroon" title="c2">f</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><b> such that </b><br/><a NAME="E3:27"/><i><font color="Green" title="E26">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c2">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Olive" title="b1">n</font>
 <b>from </b><i><a class="ref" href="seq_1.html#S1" title="SEQ_1:sch.1">SEQ_1:sch 1</a>();<br/></i>
<b>set </b><font color="Maroon" title="c3">f1</font> = <font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1;<br/>
<a NAME="E4:27"/><i><font color="Green" title="E27">A3</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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" title="c4">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/>

<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> = 
<font color="Maroon" title="c2">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">SEQM_3:def 9</a></i>
<br/>.= 
 <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="fib_num.html#E3:27"><i><font color="Green" title="E26">A2</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E2:27_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:27"/><i><font color="Green" title="E28">A4</font></i>: 
<font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c2">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" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c1">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">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" title="c4">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font></span><br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
<b>by </b><i><a class="ref" href="fib_num.html#T11" target="_self" title="FIB_NUM:th.11">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
<b>by </b><i><a class="txt" href="fib_num.html#E4:27"><i><font color="Green" title="E27">A3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Maroon" title="c4">n</font></span>)</span>
<b>by </b><i><a class="txt" href="fib_num.html#E3:27"><i><font color="Green" title="E26">A2</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E2:27_2_1"/>
<font color="Maroon" title="c1">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="fib_num.html#E1:27"><i><font color="Green" title="E25">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_2"/>
<font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> $1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span>;<br/>
<a NAME="E6:27"/>
 ex <font color="Olive" title="b1">g</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> st <br/>for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b1">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">n</font>)
 
<b>from </b><i><a class="ref" href="seq_1.html#S1" title="SEQ_1:sch.1">SEQ_1:sch 1</a>();<br/></i>
<b>then consider </b><font color="Maroon" title="c4">g</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><b> such that </b><br/><a NAME="E8:27"/><i><font color="Green" title="E29">A5</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c4">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> $1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span>;<br/>
<a NAME="E9:27"/>
 ex <font color="Olive" title="b1">h</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> st <br/>for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b1">h</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">n</font>)
 
<b>from </b><i><a class="ref" href="seq_1.html#S1" title="SEQ_1:sch.1">SEQ_1:sch 1</a>();<br/></i>
<b>then consider </b><font color="Maroon" title="c5">h</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><b> such that </b><br/><a NAME="E11:27"/><i><font color="Green" title="E30">A6</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c5">h</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">n</font>)
 ;<br/>
<a NAME="E12:27"/><i><font color="Green" title="E31">A7</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">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" title="E32">A8</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c2">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">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" title="c6">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span><br/>

<font color="Maroon" title="c2">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">n</font> = 
 <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Maroon" title="c6">n</font>
<b>by </b><i><a class="txt" href="fib_num.html#E3:27"><i><font color="Green" title="E26">A2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span>
<b>by </b><i><a class="ref" href="fib_num.html#T7" target="_self" title="FIB_NUM:th.7">Th7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T121" title="XCMPLX_1:th.121">XCMPLX_1:121</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fib_num.html#E8:27"><i><font color="Green" title="E29">A5</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
<b>by </b><i><a class="txt" href="fib_num.html#E11:27"><i><font color="Green" title="E30">A6</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E2:27_3_1"/>
<font color="Maroon" title="c2">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:27_3"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c4">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">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" title="c6">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span><br/>

<a NAME="E1:27_3_2"/>
<font color="Maroon" title="c2">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="fib_num.html#E1:27_3"><i><font color="Green" title="E32">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:27_3_2"/>
<font color="Maroon" title="c4">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:27_3"/>
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c5">h</font>
 <b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E13:27"/><i><font color="Green" title="E32">A9</font></i>: 
( <font color="Maroon" title="c5">h</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c2">f</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<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" title="E33">A10</font></i>: 
<font color="Maroon" title="c5">h</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">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" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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" title="c6">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1</span><br/>

<a NAME="E1:27_4_1_1"/>
<font color="Maroon" title="c5">h</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fib_num.html#K2" title="FIB_NUM:func.2">tau_bar</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span>
 
<b>by </b><i><a class="txt" href="fib_num.html#E11:27"><i><font color="Green" title="E30">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:27_4_1_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <b>by </b><i><a class="txt" href="fib_num.html#E27"><i><font color="Green" title="E22">Lm16</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_4_1"/>
<font color="Maroon" title="c5">h</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 <b>by </b><i><a class="ref" href="seq_2.html#T15" title="SEQ_2:th.15">SEQ_2:15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:27_4"/><i><font color="Green" title="E34">A11</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a>  is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<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" title="E35">A12</font></i>: 
for <font color="Olive" title="b1">x</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> holds <br/> ex <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">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" title="c6">x</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">x</font> implies  ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">x</font> )</span><br/>

<b>assume </b><a NAME="E1:27_4_2_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">x</font></span><br/>

<b>then consider </b><font color="Maroon" title="c7">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:27_4_2_1"/><i><font color="Green" title="E36">A13</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E4:27_4_2_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c7">k</font>
 <b>and </b><br/><a NAME="E5:27_4_2_1"/><i><font color="Green" title="E37">A14</font></i>: 
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="ref" href="fib_num.html#T3" target="_self" title="FIB_NUM:th.3">Th3</a></i>;<br/>
<a NAME="E6:27_4_2_1"/>
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">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" title="c8">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</font> implies  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">x</font> )</span><br/>

<b>assume </b><a NAME="E1:27_4_2_1_1"/>
<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">x</font></span><br/>

<a NAME="E2:27_4_2_1_1"/><b>then </b><i><font color="Green" title="E38">A15</font></i>: 
 <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Maroon" title="c8">m</font>
 
<b>by </b><i><a class="txt" href="fib_num.html#E9"><i><font color="Green" title="E8">Lm5</font></i></a></i>;<br/>
<a NAME="E3:27_4_2_1_1"/>
<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E4:27_4_2_1_1"/><b>then </b>
 <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="fib_num.html#E7"><i><font color="Green" title="E6">Lm3</font></i></a></i>;<br/>
<a NAME="E5:27_4_2_1_1"/><b>then </b>
<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Maroon" title="c8">m</font>
 
<b>by </b><i><a class="txt" href="fib_num.html#E2:27_4_2_1_1"><i><font color="Green" title="E38">A15</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:27_4_2_1_1"/><b>then </b><i><font color="Green" title="E39">A16</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">m</font>
 
<b>by </b><i><a class="txt" href="fib_num.html#E3:27"><i><font color="Green" title="E26">A2</font></i></a></i>;<br/>
<a NAME="E7:27_4_2_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T5" title="NAT_1:th.5">NAT_1:5</a></i>;<br/>
<a NAME="E8:27_4_2_1_1"/><b>then </b><i><font color="Green" title="E40">A17</font></i>: 
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="fib_num.html#E6:27_4_2_1_1"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="real_2.html#T152" title="REAL_2:th.152">REAL_2:152</a></i>;<br/>
<a NAME="E9:27_4_2_1_1"/>
<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E10:27_4_2_1_1"/><b>then </b><i><font color="Green" title="E41">A18</font></i>: 
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c7">k</font>
 
<b>by </b><i><a class="txt" href="fib_num.html#E3:27_4_2_1"><i><font color="Green" title="E36">A13</font></i></a>, <a class="ref" href="real_2.html#T151" title="REAL_2:th.151">REAL_2:151</a></i>;<br/>
<a NAME="E11:27_4_2_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">m</font>
 
<b>by </b><i><a class="txt" href="fib_num.html#E6:27_4_2_1_1"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="nat_1.html#T5" title="NAT_1:th.5">NAT_1:5</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:27_4_2_1_1"/><b>then </b><i><font color="Green" title="E42">A19</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T124" title="XREAL_1:th.124">XREAL_1:124</a></i>;<br/>
<i><font color="Green" title="E43">A20</font></i>:  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> = 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#T10" title="VALUED_1:th.10">VALUED_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> 
<b>by </b><i><a class="txt" href="fib_num.html#E12:27_4_2_1_1"><i><font color="Green" title="E42">A19</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">m</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217" title="XCMPLX_1:th.217">XCMPLX_1:217</a></i>
;<br/>
<a NAME="E14:27_4_2_1_1"/>
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c7">k</font>
 
<b>by </b><i><a class="txt" href="fib_num.html#E8:27_4_2_1_1"><i><font color="Green" title="E40">A17</font></i></a>, <a class="txt" href="fib_num.html#E10:27_4_2_1_1"><i><font color="Green" title="E41">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E15:27_4_2_1_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="fib_num.html#E5:27_4_2_1"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="fib_num.html#E13:27_4_2_1_1"><i><font color="Green" title="E43">A20</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:27_4_2_1"/>
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_4_2"/>
<font color="Maroon" title="c2">f</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a>  is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_2.html#D6" title="SEQ_2:def.6">SEQ_2:def 6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<b>hence </b><a NAME="E3:27_4_2"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="fib_num.html#E1:27_4_2"><i><font color="Green" title="E35">A12</font></i></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:27_4"/>
<font color="Maroon" title="c5">h</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c2">f</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="fib_num.html#E1:27_4"><i><font color="Green" title="E33">A10</font></i></a>, <a class="ref" href="seq_2.html#T39" title="SEQ_2:th.39">SEQ_2:39</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<b>thus </b><a NAME="E4:27_4"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="fib_num.html#E1:27_4"><i><font color="Green" title="E33">A10</font></i></a>, <a class="txt" href="fib_num.html#E2:27_4"><i><font color="Green" title="E34">A11</font></i></a>, <a class="ref" href="seq_2.html#T40" title="SEQ_2:th.40">SEQ_2:40</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c6">f2</font> = <font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2;<br/>
<a NAME="E14:27"/><i><font color="Green" title="E33">A21</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c2">f</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="E34">A22</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<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" title="c7">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">n</font> = 
<font color="Maroon" title="c2">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">SEQM_3:def 9</a></i>
<br/>.= 
 <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="fib_num.html#E3:27"><i><font color="Green" title="E26">A2</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E2:27_5_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="fib_num.html#E7"><i><font color="Green" title="E6">Lm3</font></i></a>, <a class="ref" href="nat_1.html#T5" title="NAT_1:th.5">NAT_1:5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:27_5"/><i><font color="Green" title="E35">A23</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c7">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="fib_num.html#T11" target="_self" title="FIB_NUM:th.11">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217" title="XCMPLX_1:th.217">XCMPLX_1:217</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="fib_num.html#E1:27_5"><i><font color="Green" title="E34">A22</font></i></a>, <a class="ref" href="xcmplx_1.html#T107" title="XCMPLX_1:th.107">XCMPLX_1:107</a></i>
;<br/>
<b>hence </b><a NAME="E2:27_5_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:27_5"/><b>then </b><i><font color="Green" title="E36">A24</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D6" title="SEQM_3:def.6">SEQM_3:def 6</a></i>;<br/>
<a NAME="E4:27_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="fib_num.html#E2:27_5"><i><font color="Green" title="E35">A23</font></i></a></i>;<br/>
<a NAME="E5:27_5"/><b>then </b><i><font color="Green" title="E37">A25</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="fib_num.html#E3:27_5"><i><font color="Green" title="E36">A24</font></i></a>, <a class="ref" href="seq_4.html#T40" title="SEQ_4:th.40">SEQ_4:40</a></i>;<br/>
<a NAME="E6:27_5"/><i><font color="Green" title="E38">A26</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="seqm_3.html#T43" title="SEQM_3:th.43">SEQM_3:43</a></i>;<br/>
<b>hence </b><a NAME="E7:27_5"/>
<font color="Maroon" title="c2">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c2">f</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="fib_num.html#E3:27_5"><i><font color="Green" title="E36">A24</font></i></a>, <a class="ref" href="seq_4.html#T35" title="SEQ_4:th.35">SEQ_4:35</a>, <a class="ref" href="seq_4.html#T39" title="SEQ_4:th.39">SEQ_4:39</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<b>thus </b><a NAME="E8:27_5"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="fib_num.html#E3:27_5"><i><font color="Green" title="E36">A24</font></i></a>, <a class="txt" href="fib_num.html#E5:27_5"><i><font color="Green" title="E37">A25</font></i></a>, <a class="txt" href="fib_num.html#E6:27_5"><i><font color="Green" title="E38">A26</font></i></a>, <a class="ref" href="seq_4.html#T36" title="SEQ_4:th.36">SEQ_4:36</a>, <a class="ref" href="seq_4.html#T39" title="SEQ_4:th.39">SEQ_4:39</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E15:27"/><i><font color="Green" title="E34">A27</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="fib_num.html#E12:27"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="seq_1.html#T57" title="SEQ_1:th.57">SEQ_1:57</a></i>;<br/>
<a NAME="E16:27"/><b>then </b><i><font color="Green" title="E35">A28</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c2">f</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="fib_num.html#E13:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="fib_num.html#E14:27"><i><font color="Green" title="E33">A21</font></i></a>, <a class="ref" href="seq_2.html#T19" title="SEQ_2:th.19">SEQ_2:19</a></i>;<br/>
<i><font color="Green" title="E36">A29</font></i>:  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c2">f</font></span>)</span> = 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="fib_num.html#E13:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="fib_num.html#E14:27"><i><font color="Green" title="E33">A21</font></i></a>, <a class="txt" href="fib_num.html#E15:27"><i><font color="Green" title="E34">A27</font></i></a>, <a class="ref" href="seq_2.html#T20" title="SEQ_2:th.20">SEQ_2:20</a></i>
<br/>.= 
1

;<br/>
<b>set </b><font color="Maroon" title="c7">g1</font> = <font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1;<br/>
<a NAME="E18:27"/><i><font color="Green" title="E37">A30</font></i>: 
<font color="Maroon" title="c4">g</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">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" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c4">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<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" title="c8">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<i><font color="Green" title="E38">A31</font></i>: <font color="Maroon" title="c4">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</font> = 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span>
<b>by </b><i><a class="txt" href="fib_num.html#E8:27"><i><font color="Green" title="E29">A5</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></i>
;<br/>
<a NAME="E2:27_7_1"/><i><font color="Green" title="E39">A32</font></i>: 
<a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="fib_num.html#E22"><i><font color="Green" title="E17">Lm12</font></i></a>, <a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></i>;<br/>
<a NAME="E3:27_7_1"/>
<span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="square_1.html#T85" title="SQUARE_1:th.85">SQUARE_1:85</a>, <a class="ref" href="square_1.html#T95" title="SQUARE_1:th.95">SQUARE_1:95</a>, <a class="ref" href="xcmplx_1.html#T203" title="XCMPLX_1:th.203">XCMPLX_1:203</a></i>;<br/>
<b>hence </b><a NAME="E4:27_7_1"/>
<font color="Maroon" title="c4">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="fib_num.html#E1:27_7_1"><i><font color="Green" title="E38">A31</font></i></a>, <a class="txt" href="fib_num.html#E2:27_7_1"><i><font color="Green" title="E39">A32</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_7"/>
<font color="Maroon" title="c4">g</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a>
 <b>by </b><i><a class="ref" href="seq_1.html#T7" title="SEQ_1:th.7">SEQ_1:7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E19:27"/><i><font color="Green" title="E38">A33</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c4">g</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fib_num.html#K1" title="FIB_NUM:func.1">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" title="E39">A34</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fib_num.html#K1" title="FIB_NUM:func.1">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" title="c8">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> </span><br/>

<i><font color="Green" title="E40">A35</font></i>: <font color="Maroon" title="c4">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</font> = 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span>
<b>by </b><i><a class="txt" href="fib_num.html#E8:27"><i><font color="Green" title="E29">A5</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></i>
;<br/>
<i><font color="Green" title="E41">A36</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</font> = 
<font color="Maroon" title="c4">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">SEQM_3:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span>
<b>by </b><i><a class="txt" href="fib_num.html#E8:27"><i><font color="Green" title="E29">A5</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 5</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
<br/>.= 
<a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</font></span>)</span>
<b>by </b><i><a class="txt" href="fib_num.html#E1:27_8_1"><i><font color="Green" title="E40">A35</font></i></a></i>
;<br/>
<a NAME="E3:27_8_1"/><i><font color="Green" title="E42">A37</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="fib_num.html#E18:27"><i><font color="Green" title="E37">A30</font></i></a>, <a class="ref" href="seq_1.html#T7" title="SEQ_1:th.7">SEQ_1:7</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</font> = 
<span class="p1">(<span class="default"><a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="fib_num.html#E2:27_8_1"><i><font color="Green" title="E41">A36</font></i></a>, <a class="ref" href="fib_num.html#T11" target="_self" title="FIB_NUM:th.11">Th11</a></i>
<br/>.= 
<a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span></span>)</span>

<br/>.= 
<a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
<b>by </b><i><a class="txt" href="fib_num.html#E3:27_8_1"><i><font color="Green" title="E42">A37</font></i></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>
<br/>.= 
 <a href="fib_num.html#K1" title="FIB_NUM:func.1">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" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:27_8"/><b>then </b><i><font color="Green" title="E40">A38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c4">g</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D6" title="SEQM_3:def.6">SEQM_3:def 6</a></i>;<br/>
<b>hence </b><a NAME="E3:27_8"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c4">g</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_4.html#T39" title="SEQ_4:th.39">SEQ_4:39</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> </span><br/>

<a NAME="E4:27_8"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> 
 
<b>by </b><i><a class="txt" href="fib_num.html#E1:27_8"><i><font color="Green" title="E39">A34</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:27_8"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> 
 <b>by </b><i><a class="txt" href="fib_num.html#E2:27_8"><i><font color="Green" title="E40">A38</font></i></a>, <a class="ref" href="seq_4.html#T40" title="SEQ_4:th.40">SEQ_4:40</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E20:27"/><i><font color="Green" title="E39">A39</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1 is <a href="relat_1.html#V2" title="RELAT_1:attr.2">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" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<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" title="c8">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</font> = 
<font color="Maroon" title="c2">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">SEQM_3:def 9</a></i>
<br/>.= 
 <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="fib_num.html#E3:27"><i><font color="Green" title="E26">A2</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E2:27_9_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="fib_num.html#E10"><i><font color="Green" title="E9">Lm6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_9"/>
<font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1 is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a>
 <b>by </b><i><a class="ref" href="seq_1.html#T7" title="SEQ_1:th.7">SEQ_1:7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E21:27"/><i><font color="Green" title="E40">A40</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</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" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1 = 
<font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#T36" title="SEQM_3:th.36">SEQM_3:36</a></i>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2

;<br/>
<b>hence </b><a NAME="E2:27_10"/>
<font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c8">g2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1;<br/>
<a NAME="E22:27"/><i><font color="Green" title="E41">A41</font></i>: 
( <font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1 is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1 is <a href="relat_1.html#V2" title="RELAT_1:attr.2">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" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1 is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a>
 <b>by </b><i><a class="txt" href="fib_num.html#E18:27"><i><font color="Green" title="E37">A30</font></i></a>, <a class="ref" href="seqm_3.html#T40" title="SEQM_3:th.40">SEQM_3:40</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1 is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a></span><br/>

<b>hence </b><a NAME="E2:27_11"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1 is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#T40" title="SEQM_3:th.40">SEQM_3:40</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E23:27"/><i><font color="Green" title="E42">A42</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2 is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a>
 
<b>by </b><i><a class="txt" href="fib_num.html#E20:27"><i><font color="Green" title="E39">A39</font></i></a>, <a class="txt" href="fib_num.html#E21:27"><i><font color="Green" title="E40">A40</font></i></a>, <a class="ref" href="seqm_3.html#T40" title="SEQM_3:th.40">SEQM_3:40</a></i>;<br/>
<a NAME="E24:27"/><i><font color="Green" title="E43">A43</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fib_num.html#K1" title="FIB_NUM:func.1">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" title="E44">A44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="fib_num.html#E22:27"><i><font color="Green" title="E41">A41</font></i></a>, <a class="ref" href="fib_num.html#T10" target="_self" title="FIB_NUM:th.10">Th10</a></i>;<br/>
<a NAME="E2:27_12"/><i><font color="Green" title="E45">A45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</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" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1 = 
<font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#T36" title="SEQM_3:th.36">SEQM_3:36</a></i>
<br/>.= 
<font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2

;<br/>
<b>hence </b><a NAME="E2:27_12_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:27_12"/><i><font color="Green" title="E46">A46</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="E47">A47</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2
 
<b>by </b><i><a class="txt" href="fib_num.html#E2:27_12"><i><font color="Green" title="E45">A45</font></i></a>, <a class="ref" href="seqm_3.html#T43" title="SEQM_3:th.43">SEQM_3:43</a></i>;<br/>
<a NAME="E2:27_12_2"/><b>then </b><i><font color="Green" title="E48">A48</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="fib_num.html#E16:27"><i><font color="Green" title="E35">A28</font></i></a>, <a class="ref" href="seq_4.html#T33" title="SEQ_4:th.33">SEQ_4:33</a></i>;<br/>
<a NAME="E3:27_12_2"/><i><font color="Green" title="E49">A49</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="fib_num.html#E16:27"><i><font color="Green" title="E35">A28</font></i></a>, <a class="txt" href="fib_num.html#E17:27"><i><font color="Green" title="E36">A29</font></i></a>, <a class="txt" href="fib_num.html#E1:27_12_2"><i><font color="Green" title="E47">A47</font></i></a>, <a class="ref" href="seq_4.html#T33" title="SEQ_4:th.33">SEQ_4:33</a></i>;<br/>
<a NAME="E4:27_12_2"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<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" title="c9">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E1:27_12_2_1"/><i><font color="Green" title="E50">A50</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="fib_num.html#E22:27"><i><font color="Green" title="E41">A41</font></i></a>, <a class="ref" href="seq_1.html#T7" title="SEQ_1:th.7">SEQ_1:7</a></i>;<br/>
<a NAME="E2:27_12_2_1"/><i><font color="Green" title="E51">A51</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="fib_num.html#E23:27"><i><font color="Green" title="E42">A42</font></i></a>, <a class="ref" href="seq_1.html#T7" title="SEQ_1:th.7">SEQ_1:7</a>, <a class="ref" href="xcmplx_1.html#T203" title="XCMPLX_1:th.203">XCMPLX_1:203</a></i>;<br/>
<a NAME="E3:27_12_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="fib_num.html#T11" target="_self" title="FIB_NUM:th.11">Th11</a></i>;<br/>
<b>hence </b><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" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="fib_num.html#E1:27_12_2_1"><i><font color="Green" title="E50">A50</font></i></a>, <a class="txt" href="fib_num.html#E2:27_12_2_1"><i><font color="Green" title="E51">A51</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:27_12_2"/><b>then </b><i><font color="Green" title="E50">A52</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">being_not_0</a>
 
<b>by </b><i><a class="ref" href="seq_1.html#T7" title="SEQ_1:th.7">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" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span></span>)</span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a>  is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="fib_num.html#E2:27_12_2"><i><font color="Green" title="E48">A48</font></i></a>, <a class="txt" href="fib_num.html#E3:27_12_2"><i><font color="Green" title="E49">A49</font></i></a>, <a class="ref" href="seq_2.html#T35" title="SEQ_2:th.35">SEQ_2:35</a></i>;<br/>
<b>hence </b><a NAME="E7:27_12_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_1.html#T48" title="SEQ_1:th.48">SEQ_1:48</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

 <a href="seq_2.html#K2" title="SEQ_2:func.2">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" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span></span>)</span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> = 
1 <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> 
<b>by </b><i><a class="txt" href="fib_num.html#E2:27_12_2"><i><font color="Green" title="E48">A48</font></i></a>, <a class="txt" href="fib_num.html#E3:27_12_2"><i><font color="Green" title="E49">A49</font></i></a>, <a class="txt" href="fib_num.html#E5:27_12_2"><i><font color="Green" title="E50">A52</font></i></a>, <a class="ref" href="seq_2.html#T36" title="SEQ_2:th.36">SEQ_2:36</a></i>
<br/>.= 
1

;<br/>
<b>hence </b><a NAME="E9:27_12_2"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="seq_1.html#T48" title="SEQ_1:th.48">SEQ_1:48</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:27_12"/><i><font color="Green" title="E47">A53</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fib_num.html#K1" title="FIB_NUM:func.1">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" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1
 
<b>by </b><i><a class="ref" href="seqm_3.html#T43" title="SEQM_3:th.43">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" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  )
 <b>by </b><i><a class="txt" href="fib_num.html#E19:27"><i><font color="Green" title="E38">A33</font></i></a>, <a class="ref" href="seq_4.html#T33" title="SEQ_4:th.33">SEQ_4:33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:27_12"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="fib_num.html#E1:27_12"><i><font color="Green" title="E44">A44</font></i></a>, <a class="txt" href="fib_num.html#E3:27_12"><i><font color="Green" title="E46">A46</font></i></a>, <a class="ref" href="seq_2.html#T28" title="SEQ_2:th.28">SEQ_2:28</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> </span><br/>

 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span></span>)</span> = 
1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> 
<b>by </b><i><a class="txt" href="fib_num.html#E1:27_12"><i><font color="Green" title="E44">A44</font></i></a>, <a class="txt" href="fib_num.html#E3:27_12"><i><font color="Green" title="E46">A46</font></i></a>, <a class="txt" href="fib_num.html#E4:27_12"><i><font color="Green" title="E47">A53</font></i></a>, <a class="ref" href="seq_2.html#T29" title="SEQ_2:th.29">SEQ_2:29</a></i>
<br/>.= 
 <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> 

;<br/>
<b>hence </b><a NAME="E7:27_12"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E25:27"/><i><font color="Green" title="E44">A54</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1
 
<b>by </b><i><a class="ref" href="seqm_3.html#T43" title="SEQM_3:th.43">SEQM_3:43</a></i>;<br/>
<b>hence </b><a NAME="E2:27_13"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="fib_num.html#E16:27"><i><font color="Green" title="E35">A28</font></i></a>, <a class="txt" href="fib_num.html#E17:27"><i><font color="Green" title="E36">A29</font></i></a>, <a class="ref" href="seq_4.html#T33" title="SEQ_4:th.33">SEQ_4:33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E26:27"/><i><font color="Green" title="E45">A55</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fib_num.html#K1" title="FIB_NUM:func.1">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" title="E46">A56</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="fib_num.html#E22:27"><i><font color="Green" title="E41">A41</font></i></a>, <a class="ref" href="fib_num.html#T10" target="_self" title="FIB_NUM:th.10">Th10</a></i>;<br/>
<b>hence </b><a NAME="E2:27_14"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="fib_num.html#E24:27"><i><font color="Green" title="E43">A43</font></i></a>, <a class="txt" href="fib_num.html#E25:27"><i><font color="Green" title="E44">A54</font></i></a>, <a class="ref" href="seq_2.html#T28" title="SEQ_2:th.28">SEQ_2:28</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> </span><br/>

<a NAME="E3:27_14"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a>  <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
 
<b>by </b><i><a class="txt" href="fib_num.html#E24:27"><i><font color="Green" title="E43">A43</font></i></a>, <a class="txt" href="fib_num.html#E25:27"><i><font color="Green" title="E44">A54</font></i></a>, <a class="txt" href="fib_num.html#E1:27_14"><i><font color="Green" title="E46">A56</font></i></a>, <a class="ref" href="seq_2.html#T29" title="SEQ_2:th.29">SEQ_2:29</a></i>;<br/>
<b>hence </b><a NAME="E4:27_14"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E27:27"/><i><font color="Green" title="E46">A57</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 2</span>)</span> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> 1
 
<b>by </b><i><a class="txt" href="fib_num.html#E5:27"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="fib_num.html#E21:27"><i><font color="Green" title="E40">A40</font></i></a>, <a class="ref" href="seqm_3.html#T43" title="SEQM_3:th.43">SEQM_3:43</a></i>;<br/>
<b>hence </b><a NAME="E28:27"/>
<font color="Maroon" title="c1">F</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="fib_num.html#E26:27"><i><font color="Green" title="E45">A55</font></i></a>, <a class="ref" href="seq_4.html#T35" title="SEQ_4:th.35">SEQ_4:35</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> </span><br/>

<b>thus </b><a NAME="E29:27"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fib_num.html#K1" title="FIB_NUM:func.1">tau</a> 
 <b>by </b><i><a class="txt" href="fib_num.html#E26:27"><i><font color="Green" title="E45">A55</font></i></a>, <a class="txt" href="fib_num.html#E27:27"><i><font color="Green" title="E46">A57</font></i></a>, <a class="ref" href="seq_4.html#T36" title="SEQ_4:th.36">SEQ_4:36</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
