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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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> ] <b>means </b>( <font color="Maroon">P<sub>1</sub></font>[$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] &amp; <font color="Maroon">P<sub>1</sub></font>[$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2] );<br/>
<a NAME="E1:33_1"/><i><font color="Green" title="E25">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<b>by </b><i><a class="txt" href="fib_num2.html#E1:33"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="fib_num2.html#E2:33"><i><font color="Green" title="E23">A2</font></i></a></i>;<br/>
<a NAME="E2:33_1"/><i><font color="Green" title="E26">A5</font></i>: 
for <font color="Olive" title="b1">k</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">k</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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="E1:33_1_1"/>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
;<br/>
<a NAME="E2:33_1_1"/><b>then </b><i><font color="Green" title="E27">A6</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 is  non <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>  <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> 
 
<b>by </b><i><a class="ref" href="nat_2.html#D1" title="NAT_2:def.1">NAT_2:def 1</a></i>;<br/>
<b>assume </b><a NAME="E3:33_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font>]
 ;<br/>

<a NAME="E4:33_1_1"/><b>then </b>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] &amp; <font color="Maroon">P<sub>1</sub></font>[<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )
 
;<br/>
<b>hence </b><a NAME="E5:33_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="fib_num2.html#E3:33"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="fib_num2.html#E2:33_1_1"><i><font color="Green" title="E27">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:33_1"/><i><font color="Green" title="E27">A7</font></i>: 
for <font color="Olive" title="b1">k</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="binarith.html#S1" title="BINARITH:sch.1">BINARITH:sch 1</a>(<a class="txt" href="fib_num2.html#E1:33_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="fib_num2.html#E2:33_1"><i><font color="Green" title="E26">A5</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c1">k</font> be  non <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>  <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="E4:33_1"/>
( <font color="Maroon" title="c1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_2.html#D1" title="NAT_2:def.1">NAT_2:def 1</a></i>;<br/>
<a NAME="E5:33_1"/><b>then </b><i><font color="Green" title="E28">A8</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="nat_2.html#T21" title="NAT_2:th.21">NAT_2:21</a></i>;<br/>
<a NAME="E6:33_1"/><b>then </b>
<font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E7:33_1"/><b>then </b><i><font color="Green" title="E29">A9</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
;<br/>
<a NAME="E8:33_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 
<b>by </b><i><a class="txt" href="fib_num2.html#E5:33_1"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/>
<b>hence </b><a NAME="E9:33_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">k</font>]
 <b>by </b><i><a class="txt" href="fib_num2.html#E3:33_1"><i><font color="Green" title="E27">A7</font></i></a>, <a class="txt" href="fib_num2.html#E7:33_1"><i><font color="Green" title="E29">A9</font></i></a></i>;<br/>


</div>
