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

<b>assume </b><a NAME="E1:62"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; ( ( <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 ) or ( <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 &amp; <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 ) ) )
 ;<br/>

<a NAME="E2:62"/>
not <font color="Maroon">k</font> is <a href="realset1.html#V1">trivial</a>
 
<b>by </b><i><a class="ref" href="fib_num2.html#T13" target="_self">Th13</a>, <a class="ref" href="nat_2.html#D1">NAT_2:def 1</a></i>;<br/>
<a NAME="E3:62"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_2.html#T31">NAT_2:31</a></i>;<br/>
<a NAME="E4:62"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:62"/>
(  <a href="pre_ff.html#K3">Fib</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <font color="Maroon">n</font> implies <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:62_1"/><i><font color="Green">E39</font></i>: 
 <a href="pre_ff.html#K3">Fib</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <font color="Maroon">n</font>
 ;<br/>

<b>assume </b><a NAME="E2:62_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:62_1_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font> or <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_1"><b>suppose </b></a><a NAME="E1:62_1_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:62_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="fib_num2.html#T13" target="_self">Th13</a>, , <a class="ref" href="fib_num2.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2"><b>suppose </b></a><a NAME="E1:62_1_1_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:62_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="fib_num2.html#T14" target="_self">Th14</a>, , <a class="ref" href="fib_num2.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E6:62"/>
(  <a href="pre_ff.html#K3">Fib</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <font color="Maroon">n</font> iff <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 ;<br/>


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