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

<b>let </b><font color="Maroon" title="c2">it1</font>, <font color="Maroon" title="c3">it2</font> be    <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>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">K19</a>(<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default">$2 <a href="pre_ff.html#K2" title="PRE_FF:func.2">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">$2 <a href="pre_ff.html#K1" title="PRE_FF:func.1">`1</a> </span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default">$2 <a href="pre_ff.html#K2" title="PRE_FF:func.2">`2</a> </span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/>
<b>given </b><font color="Maroon" title="c4">L1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span><b> such that </b><a NAME="E1:13_1_2"/><i><font color="Green" title="E13">A3</font></i>: 
<font color="Maroon" title="c2">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_ff.html#K1" title="PRE_FF:func.1">`1</a> 
 <b>and </b><br/><a NAME="E2:13_1_2"/><i><font color="Green" title="E14">A4</font></i>: 
( <font color="Maroon" title="c4">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; ( for <font color="Olive" title="b1">n</font> being   <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" title="c4">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c4">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>) ) )
 ;<br/>

<b>given </b><font color="Maroon" title="c5">L2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span><b> such that </b><a NAME="E3:13_1_2"/><i><font color="Green" title="E15">A5</font></i>: 
<font color="Maroon" title="c3">it2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">L2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_ff.html#K1" title="PRE_FF:func.1">`1</a> 
 <b>and </b><br/><a NAME="E4:13_1_2"/><i><font color="Green" title="E16">A6</font></i>: 
( <font color="Maroon" title="c5">L2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; ( for <font color="Olive" title="b1">n</font> being   <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" title="c5">L2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c5">L2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>) ) )
 ;<br/>

<a NAME="E5:13_1_2"/>
<font color="Maroon" title="c4">L1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">L2</font>
 
<b>from </b><i><a class="ref" href="recdef_1.html#S12" title="RECDEF_1:sch.12">RECDEF_1:sch 12</a>(<a class="txt" href="fib_num3.html#E2:13_1_2"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="fib_num3.html#E4:13_1_2"><i><font color="Green" title="E16">A6</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:13_1_2"/>
<font color="Maroon" title="c2">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">it2</font>
 <b>by </b><i><a class="txt" href="fib_num3.html#E1:13_1_2"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="fib_num3.html#E3:13_1_2"><i><font color="Green" title="E15">A5</font></i></a></i>;<br/>


</div>
