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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>thus </b><a NAME="E1:21"/>
 <a href="fib_num3.html#K1" target="_self">Lucas</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="fib_num3.html#K1" target="_self">Lucas</a> <font color="Maroon">c<sub>1</sub></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="nat_1.html#NM1">Nat</a>] means  <a href="fib_num3.html#K1" target="_self">Lucas</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="fib_num3.html#K1" target="_self">Lucas</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:21_1"/><i><font color="Green">E21</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">Th11</a>, <a class="ref" href="fib_num3.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E2:21_1"/><i><font color="Green">E22</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">Th14</a>, <a class="ref" href="fib_num3.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E3:21_1"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</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">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">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">c<sub>2</sub></font>]
 <b>and </b><br/><a NAME="E2:21_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>

<b>thus </b><a NAME="E3:21_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</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" target="_self">Lucas</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="fib_num3.html#K1" target="_self">Lucas</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fib_num3.html#K1" target="_self">Lucas</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 3</span>)</span>
 
<b>by </b><i><a class="ref" href="fib_num3.html#T13" target="_self">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">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 2]
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E4:21_1"/>
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="fib_num2.html#S1">FIB_NUM2:sch 1</a>(<a class="txt" href="fib_num3.html#E1:21_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="fib_num3.html#E2:21_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="fib_num3.html#E3:21_1"><i><font color="Green">E23</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:21_1"/>
 <a href="fib_num3.html#K1" target="_self">Lucas</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="fib_num3.html#K1" target="_self">Lucas</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


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


</div>
