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<b>let </b><font color="Maroon" title="c1">m</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/>

<b>thus </b><a NAME="E1:21"/>
 <a href="fib_num3.html#K1" title="FIB_NUM3:func.1">Lucas</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="fib_num3.html#K1" title="FIB_NUM3:func.1">Lucas</a> <font color="Maroon" title="c1">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><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> <a href="fib_num3.html#K1" title="FIB_NUM3:func.1">Lucas</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="fib_num3.html#K1" title="FIB_NUM3:func.1">Lucas</a> $1;<br/>
<a NAME="E1:21_1"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<b>by </b><i><a class="ref" href="fib_num3.html#T11" target="_self" title="FIB_NUM3:th.11">Th11</a>, <a class="ref" href="fib_num3.html#T14" target="_self" title="FIB_NUM3:th.14">Th14</a></i>;<br/>
<a NAME="E2:21_1"/><i><font color="Green" title="E22">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[2]
 
<b>by </b><i><a class="ref" href="fib_num3.html#T14" target="_self" title="FIB_NUM3:th.14">Th14</a>, <a class="ref" href="fib_num3.html#T15" target="_self" title="FIB_NUM3:th.15">Th15</a></i>;<br/>
<a NAME="E3:21_1"/><i><font color="Green" title="E23">A3</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>] &amp; <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] 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> 2]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">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/>

<b>assume </b><b>that </b><br/><a NAME="E1:21_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font>]
 <b>and </b><br/><a NAME="E2:21_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ;<br/>

<b>thus </b><a NAME="E3:21_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:21_1_1_1"/>
<span class="p1">(<span class="default"><a href="fib_num3.html#K1" title="FIB_NUM3:func.1">Lucas</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="fib_num3.html#K1" title="FIB_NUM3:func.1">Lucas</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fib_num3.html#K1" title="FIB_NUM3:func.1">Lucas</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span>
 
<b>by </b><i><a class="ref" href="fib_num3.html#T13" target="_self" title="FIB_NUM3:th.13">Th13</a></i>;<br/>
<b>hence </b><a NAME="E2:21_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2]
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E4:21_1"/>
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="fib_num2.html#S1" title="FIB_NUM2:sch.1">FIB_NUM2:sch 1</a>(<a class="txt" href="fib_num3.html#E1:21_1"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="fib_num3.html#E2:21_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="fib_num3.html#E3:21_1"><i><font color="Green" title="E23">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:21_1"/>
 <a href="fib_num3.html#K1" title="FIB_NUM3:func.1">Lucas</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="fib_num3.html#K1" title="FIB_NUM3:func.1">Lucas</a> <font color="Maroon" title="c1">m</font>
 ;<br/>


</div><b>end;</b></div>


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