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<b>let </b><font color="Maroon" title="c1">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:61"/>
(  not  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_1"/><i><font color="Green" title="E46">A1</font></i>: 
 <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>

<b>assume </b><a NAME="E2:61_1"/>
( not <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; not <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ;<br/>

<a NAME="E3:61_1"/><b>then </b>
( ( <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 ) &amp; ( <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 or <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 ) )
 
<b>by </b><i><a class="ref" href="nat_2.html#T21" title="NAT_2:th.21">NAT_2:21</a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:61_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fib_num2.html#E1:61_1"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="fib_num2.html#T23" target="_self" title="FIB_NUM2:th.23">Th23</a>, <a class="ref" href="fib_num2.html#T48" target="_self" title="FIB_NUM2:th.48">Th48</a>, <a class="ref" href="pre_ff.html#T1" title="PRE_FF:th.1">PRE_FF:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:61"/>
(  <a href="pre_ff.html#K3" title="PRE_FF:func.3">Fib</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff ( <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )
 <b>by </b><i><a class="ref" href="fib_num2.html#T23" target="_self" title="FIB_NUM2:th.23">Th23</a>, <a class="ref" href="pre_ff.html#T1" title="PRE_FF:th.1">PRE_FF:1</a></i>;<br/>


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