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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ,   <a href="subset_1.html#M2">Element</a> of <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="mcart_1.html#K12">:]</a></span>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="pre_ff.html#K2" target="_self">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="pre_ff.html#K1" target="_self">`1</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="pre_ff.html#K2" target="_self">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>;<br/>
<b>consider </b><font color="Maroon">fib</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="mcart_1.html#K12">:]</a></span><b> such that </b><br/><a NAME="E2:3"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">fib</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,1</span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E3:3"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">fib</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>,<font color="Maroon">fib</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="recdef_1.html#S4">RECDEF_1:sch 4</a>();<br/></i>
<b>thus </b> <a href="pre_ff.html#K3" target="_self">Fib</a> 0 = 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,1</span><a href="domain_1.html#K1">]</a></span> <a href="pre_ff.html#K1" target="_self">`1</a> 
<b>by </b><i>, , </i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>

<b>thus </b> <a href="pre_ff.html#K3" target="_self">Fib</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">fib</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="pre_ff.html#K1" target="_self">`1</a> 
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fib</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="pre_ff.html#K2" target="_self">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">fib</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="pre_ff.html#K1" target="_self">`1</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">fib</font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="pre_ff.html#K2" target="_self">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="pre_ff.html#K1" target="_self">`1</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,1</span><a href="domain_1.html#K1">]</a></span> <a href="pre_ff.html#K2" target="_self">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<i><font color="Green">E36</font></i>: <span class="p1">(<span class="default"><font color="Maroon">fib</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="pre_ff.html#K1" target="_self">`1</a>  = 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fib</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="pre_ff.html#K2" target="_self">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">fib</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="pre_ff.html#K1" target="_self">`1</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">fib</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="pre_ff.html#K2" target="_self">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="pre_ff.html#K1" target="_self">`1</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">fib</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="pre_ff.html#K2" target="_self">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>
<b>thus </b> <a href="pre_ff.html#K3" target="_self">Fib</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">fib</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="pre_ff.html#K1" target="_self">`1</a> 
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fib</font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="pre_ff.html#K2" target="_self">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">fib</font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="pre_ff.html#K1" target="_self">`1</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">fib</font> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="pre_ff.html#K2" target="_self">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="pre_ff.html#K1" target="_self">`1</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">fib</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="pre_ff.html#K2" target="_self">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fib</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="pre_ff.html#K2" target="_self">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">fib</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="pre_ff.html#K1" target="_self">`1</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">fib</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="pre_ff.html#K2" target="_self">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="pre_ff.html#K2" target="_self">`2</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fib</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="pre_ff.html#K1" target="_self">`1</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fib</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="pre_ff.html#K1" target="_self">`1</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_ff.html#K3" target="_self">Fib</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fib</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="pre_ff.html#K1" target="_self">`1</a> </span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_ff.html#K3" target="_self">Fib</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="pre_ff.html#K3" target="_self">Fib</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, , </i>
;<br/>


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