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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">a<sub>2</sub></font></span>)</span>;<br/>
<a NAME="E1:12"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">m</font>, <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>,<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">m</font>]
 
;<br/>
<a NAME="E2:12"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st ( for <font color="Olive">m</font>, <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>,<font color="Olive">n</font>] ) holds <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">m</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:12_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">m</font>, <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>,<font color="Olive">n</font>]
 ;<br/>

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E2:12_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_1_1"/>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> or <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> )
 </span><b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1"><b>suppose </b></a><a NAME="E1:12_1_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:12_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">m</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2"><b>suppose </b></a><a NAME="E1:12_1_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:12_1_1_2"/><i><font color="Green">E34</font></i>: 
not <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><b>thus </b><a NAME="E3:12_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">m</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_1_1_2_1_1"/>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span><b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:12_1_1_2_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:12_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_ff.html#K3">Fib</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="newton.html#T65">NEWTON:65</a>, <a class="ref" href="pre_ff.html#T1">PRE_FF:1</a></i>;<br/><b>hence </b><a NAME="E3:12_1_1_2_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">m</font>]
 <b>by </b><i>, <a class="ref" href="newton.html#T65">NEWTON:65</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:12_1_1_2_1_1_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>thus </b><a NAME="E2:12_1_1_2_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">m</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">r</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E2:12_1_1_2_1_1_2_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E4:12_1_1_2_1_1_2_1"/>
<font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">rr</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:12_1_1_2_1_1_2_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rr</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">rr</font> = <font color="Maroon">rr</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E8:12_1_1_2_1_1_2_1"/>
 <a href="pre_ff.html#K3">Fib</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <span class="p3">(<span class="default"><font color="Maroon">rr</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <font color="Maroon">rr</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E9:12_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="euler_1.html#T9">EULER_1:9</a></i>;<br/>
<a NAME="E10:12_1_1_2_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E11:12_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3">Fib</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:12_1_1_2_1_1_2_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E13:12_1_1_2_1_1_2_1"/>
<font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:12_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E15:12_1_1_2_1_1_2_1"/>
<font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">r</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E16:12_1_1_2_1_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">m</font>]
 <b>by </b><i>, , , </i>;<br/>


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

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

</div><b>end;</b></div>
<b>thus </b><a NAME="E3:12"/>
for <font color="Olive">m</font>, <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>,<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="fib_num.html#S2" target="_self">FIB_NUM:sch 2</a>(, );</i><br/>


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