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

<b>set </b><font color="Maroon" title="c1">r04</font> = <font color="Maroon">F<sub>4</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>());<br/>
<b>set </b><font color="Maroon" title="c2">r14</font> = <font color="Maroon">F<sub>4</sub></font>(1,<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()));<br/>
<b>set </b><font color="Maroon" title="c3">r24</font> = <font color="Maroon">F<sub>4</sub></font>(2,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()),<font color="Maroon">F<sub>4</sub></font>(1,<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>())));<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default">$2 <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a> </span>)</span>,<span class="p2">(<span class="default">$2 <a href="recdef_2.html#K6" title="RECDEF_2:func.6">`3_4</a> </span>)</span>,<span class="p2">(<span class="default">$2 <a href="recdef_2.html#K7" title="RECDEF_2:func.7">`4_4</a> </span>)</span>,<font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default">$2 <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a> </span>)</span>,<span class="p1">(<span class="default">$2 <a href="recdef_2.html#K6" title="RECDEF_2:func.6">`3_4</a> </span>)</span>,<span class="p1">(<span class="default">$2 <a href="recdef_2.html#K7" title="RECDEF_2:func.7">`4_4</a> </span>)</span>)</span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon">H<sub>1</sub></font>($1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2,$2);<br/>
<b>consider </b><font color="Maroon" title="c4">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E2:20_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>and </b><br/><a NAME="E3:20_1"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()),<font color="Maroon">F<sub>4</sub></font>(1,<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>())),<font color="Maroon">F<sub>4</sub></font>(2,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()),<font color="Maroon">F<sub>4</sub></font>(1,<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>())))</span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>
 <b>and </b><br/><a NAME="E4:20_1"/><i><font color="Green" title="E17">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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>2</sub></font>(<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="recdef_1.html#S3" title="RECDEF_1:sch.3">RECDEF_1:sch 3</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> $1,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>())</span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()),<font color="Maroon">F<sub>4</sub></font>(1,<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()))</span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> $1,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span>;<br/>
<a NAME="E5:20_1"/><i><font color="Green" title="E18">A3</font></i>: 
for <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">c</font>] implies  not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">c</font>] ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">c</font>] implies  not <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">c</font>] ) &amp; ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">c</font>] implies  not <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">c</font>] ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:20_1_1"/>
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/>

<a NAME="E2:20_1_1"/><b>then </b>
 <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <font color="Maroon" title="c5">c</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font>
 
<b>by </b><i><a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:20_1_1"/>
( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">c</font>] implies  not <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">c</font>] ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">c</font>] implies  not <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c5">c</font>] ) &amp; ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">c</font>] implies  not <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c5">c</font>] ) )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:20_1"/><i><font color="Green" title="E19">A4</font></i>: 
for <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  or <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">c</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">c</font>] or <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">c</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:20_1_2"/>
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/>

<a NAME="E2:20_1_2"/><b>then </b>
 <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <font color="Maroon" title="c5">c</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font>
 
<b>by </b><i><a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:20_1_2"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">c</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">c</font>] or <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c5">c</font>] )
 <b>by </b><i><a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c5">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E8:20_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>and </b><br/><a NAME="E9:20_1"/><i><font color="Green" title="E20">A5</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] implies <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">x</font>) ) &amp; ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>] implies <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">x</font>) ) &amp; ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>] implies <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">x</font>) ) )
 <b>from </b><i><a class="ref" href="recdef_2.html#S1" target="_self" title="RECDEF_2:sch.1">RECDEF_2:sch 1</a>(<a class="txt" href="recdef_2.html#E5:20_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="recdef_2.html#E6:20_1"><i><font color="Green" title="E19">A4</font></i></a>);<br/></i>
<a NAME="E10:20_1"/><i><font color="Green" title="E21">A6</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>())</span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="recdef_2.html#E9:20_1"><i><font color="Green" title="E20">A5</font></i></a></i>;<br/>
<a NAME="E11:20_1"/><i><font color="Green" title="E22">A7</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()),<font color="Maroon">F<sub>4</sub></font>(1,<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()))</span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="recdef_2.html#E9:20_1"><i><font color="Green" title="E20">A5</font></i></a></i>;<br/>
<a NAME="E12:20_1"/><i><font color="Green" title="E23">A8</font></i>: 
 <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> 2,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E24">A9</font></i>: <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 = 
<font color="Maroon">H<sub>5</sub></font>(2)
<b>by </b><i><a class="txt" href="recdef_2.html#E9:20_1"><i><font color="Green" title="E20">A5</font></i></a></i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()),<font color="Maroon">F<sub>4</sub></font>(1,<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>())),<font color="Maroon">F<sub>4</sub></font>(2,<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()),<font color="Maroon">F<sub>4</sub></font>(1,<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>())))</span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>
<b>by </b><i><a class="txt" href="recdef_2.html#E3:20_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="recdef_2.html#E12:20_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>
;<br/>
<a NAME="E14:20_1"/><i><font color="Green" title="E25">A10</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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> 3</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> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2,<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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> 2</span>)</span>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:20_1_4"/><i><font color="Green" title="E26">A11</font></i>: 
(  <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 &amp;  <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3 )
 
<b>by </b><i><a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></i>;<br/>
<a NAME="E2:20_1_4"/><i><font color="Green" title="E27">A12</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/>
<a NAME="E3:20_1_4"/><b>then </b>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<b>then </b><i><font color="Green" title="E28">A13</font></i>: <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> = 
<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2)
<b>by </b><i><a class="txt" href="recdef_2.html#E9:20_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="recdef_2.html#E1:20_1_4"><i><font color="Green" title="E26">A11</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>
<b>by </b><i><a class="txt" href="recdef_2.html#E1:20_1_4"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>
;<br/>
<a NAME="E5:20_1_4"/><i><font color="Green" title="E29">A14</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">3 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span>
 
;<br/>
<a NAME="E6:20_1_4"/><i><font color="Green" title="E30">A15</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/>
<a NAME="E7:20_1_4"/>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3
 
<b>by </b><i><a class="txt" href="recdef_2.html#E2:20_1_4"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span> = 
<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3)
<b>by </b><i><a class="txt" href="recdef_2.html#E9:20_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="recdef_2.html#E1:20_1_4"><i><font color="Green" title="E26">A11</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="recdef_2.html#E1:20_1_4"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="recdef_2.html#E5:20_1_4"><i><font color="Green" title="E29">A14</font></i></a>, <a class="txt" href="recdef_2.html#E6:20_1_4"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2,<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>)
<b>by </b><i><a class="txt" href="recdef_2.html#E4:20_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="recdef_2.html#E4:20_1_4"><i><font color="Green" title="E28">A13</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>6</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</a> ;<br/>
<b>consider </b><font color="Maroon" title="c6">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E16:20_1"/><i><font color="Green" title="E26">A16</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>and </b><br/><a NAME="E17:20_1"/><i><font color="Green" title="E27">A17</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>6</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S4" title="FUNCT_1:sch.4">FUNCT_1:sch 4</a>();<br/></i>
<b>take </b>
<font color="Maroon" title="c6">f</font>
;<br/>

<b>thus </b><a NAME="E18:20_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="recdef_2.html#E16:20_1"><i><font color="Green" title="E26">A16</font></i></a></i>;<br/>

<b>thus </b><i><font color="Green" title="E28">A18</font></i>: <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0</span>)</span> <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E17:20_1"><i><font color="Green" title="E27">A17</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>1</sub></font>()
<b>by </b><i><a class="txt" href="recdef_2.html#E10:20_1"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="recdef_2.html#D4" target="_self" title="RECDEF_2:def.4">Def4</a></i>
;<br/>

<b>thus </b><i><font color="Green" title="E29">A19</font></i>: <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E17:20_1"><i><font color="Green" title="E27">A17</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>2</sub></font>()
<b>by </b><i><a class="txt" href="recdef_2.html#E11:20_1"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="recdef_2.html#D4" target="_self" title="RECDEF_2:def.4">Def4</a></i>
;<br/>

<b>thus </b><i><font color="Green" title="E30">A20</font></i>: <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span> <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E17:20_1"><i><font color="Green" title="E27">A17</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>()
<b>by </b><i><a class="txt" href="recdef_2.html#E13:20_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="recdef_2.html#D4" target="_self" title="RECDEF_2:def.4">Def4</a></i>
;<br/>

<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="recdef_2.html#K6" title="RECDEF_2:func.6">`3_4</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="recdef_2.html#K7" title="RECDEF_2:func.7">`4_4</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K6" title="RECDEF_2:func.6">`3_4</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K6" title="RECDEF_2:func.6">`3_4</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K7" title="RECDEF_2:func.7">`4_4</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K6" title="RECDEF_2:func.6">`3_4</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span></span>)</span> <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</a>  &amp; <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>($1,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span>) );<br/>
<a NAME="E22:20_1"/><i><font color="Green" title="E31">A21</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_8"><b>proof </b></a><div class="add">

<b>thus </b><i><font color="Green" title="E32">A22</font></i>: <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span></span>)</span> <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E17:20_1"><i><font color="Green" title="E27">A17</font></i></a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2,<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>) <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E14:20_1"><i><font color="Green" title="E25">A10</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D4" target="_self" title="RECDEF_2:def.4">Def4</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</a>  = 
<font color="Maroon">F<sub>2</sub></font>()
<b>by </b><i><a class="txt" href="recdef_2.html#E11:20_1"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="recdef_2.html#D4" target="_self" title="RECDEF_2:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0</span>)</span> <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E10:20_1"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="recdef_2.html#D5" target="_self" title="RECDEF_2:def.5">Def5</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0</span>)</span> <a href="recdef_2.html#K6" title="RECDEF_2:func.6">`3_4</a>  = 
<font color="Maroon">F<sub>3</sub></font>()
<b>by </b><i><a class="txt" href="recdef_2.html#E10:20_1"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="recdef_2.html#D6" target="_self" title="RECDEF_2:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E11:20_1"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="recdef_2.html#D5" target="_self" title="RECDEF_2:def.5">Def5</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0</span>)</span> <a href="recdef_2.html#K7" title="RECDEF_2:func.7">`4_4</a>  = 
<font color="Maroon">F<sub>4</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>())
<b>by </b><i><a class="txt" href="recdef_2.html#E10:20_1"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="recdef_2.html#D7" target="_self" title="RECDEF_2:def.7">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K6" title="RECDEF_2:func.6">`3_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E11:20_1"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="recdef_2.html#D6" target="_self" title="RECDEF_2:def.6">Def6</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a>  = 
<font color="Maroon">F<sub>3</sub></font>()
<b>by </b><i><a class="txt" href="recdef_2.html#E11:20_1"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="recdef_2.html#D5" target="_self" title="RECDEF_2:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E13:20_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="recdef_2.html#D4" target="_self" title="RECDEF_2:def.4">Def4</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K6" title="RECDEF_2:func.6">`3_4</a>  = 
<font color="Maroon">F<sub>4</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>())
<b>by </b><i><a class="txt" href="recdef_2.html#E11:20_1"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="recdef_2.html#D6" target="_self" title="RECDEF_2:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E13:20_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="recdef_2.html#D5" target="_self" title="RECDEF_2:def.5">Def5</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K7" title="RECDEF_2:func.7">`4_4</a>  = 
<font color="Maroon">F<sub>4</sub></font>(1,<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()))
<b>by </b><i><a class="txt" href="recdef_2.html#E11:20_1"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="recdef_2.html#D7" target="_self" title="RECDEF_2:def.7">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K6" title="RECDEF_2:func.6">`3_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E13:20_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="recdef_2.html#D6" target="_self" title="RECDEF_2:def.6">Def6</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a>  = 
<font color="Maroon">H<sub>1</sub></font>(0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2,<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>) <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D4" target="_self" title="RECDEF_2:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span></span>)</span> <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E14:20_1"><i><font color="Green" title="E25">A10</font></i></a></i>
;<br/>

<b>thus </b><a NAME="E9:20_1_8"/>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(0,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span>)
 <b>by </b><i><a class="txt" href="recdef_2.html#E13:20_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="recdef_2.html#E19:20_1"><i><font color="Green" title="E28">A18</font></i></a>, <a class="txt" href="recdef_2.html#E20:20_1"><i><font color="Green" title="E29">A19</font></i></a>, <a class="txt" href="recdef_2.html#E21:20_1"><i><font color="Green" title="E30">A20</font></i></a>, <a class="txt" href="recdef_2.html#E1:20_1_8"><i><font color="Green" title="E32">A22</font></i></a>, <a class="ref" href="recdef_2.html#D5" target="_self" title="RECDEF_2:def.5">Def5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:20_1"/><i><font color="Green" title="E32">A23</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:20_1_9"/><i><font color="Green" title="E33">A24</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c7">x</font>]
 ;<br/>

<a NAME="E2:20_1_9"/><i><font color="Green" title="E34">A25</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>
 
;<br/>
<b>thus </b><i><font color="Green" title="E35">A26</font></i>: <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span></span>)</span> <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E17:20_1"><i><font color="Green" title="E27">A17</font></i></a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2,<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>) <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E14:20_1"><i><font color="Green" title="E25">A10</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D4" target="_self" title="RECDEF_2:def.4">Def4</a></i>
;<br/>

<b>thus </b><a NAME="E4:20_1_9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a> 
 <b>by </b><i><a class="txt" href="recdef_2.html#E1:20_1_9"><i><font color="Green" title="E33">A24</font></i></a></i>;<br/>

<b>thus </b><a NAME="E5:20_1_9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K6" title="RECDEF_2:func.6">`3_4</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a> 
 <b>by </b><i><a class="txt" href="recdef_2.html#E1:20_1_9"><i><font color="Green" title="E33">A24</font></i></a></i>;<br/>

<b>thus </b><a NAME="E6:20_1_9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K7" title="RECDEF_2:func.7">`4_4</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K6" title="RECDEF_2:func.6">`3_4</a> 
 <b>by </b><i><a class="txt" href="recdef_2.html#E1:20_1_9"><i><font color="Green" title="E33">A24</font></i></a></i>;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a>  = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2,<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>) <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D4" target="_self" title="RECDEF_2:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E14:20_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="recdef_2.html#E2:20_1_9"><i><font color="Green" title="E34">A25</font></i></a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K6" title="RECDEF_2:func.6">`3_4</a>  = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2,<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>) <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D5" target="_self" title="RECDEF_2:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E14:20_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="recdef_2.html#E2:20_1_9"><i><font color="Green" title="E34">A25</font></i></a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K7" title="RECDEF_2:func.7">`4_4</a>  = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2,<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>) <a href="recdef_2.html#K6" title="RECDEF_2:func.6">`3_4</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D6" target="_self" title="RECDEF_2:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K6" title="RECDEF_2:func.6">`3_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E14:20_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="recdef_2.html#E2:20_1_9"><i><font color="Green" title="E34">A25</font></i></a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a>  = 
<font color="Maroon">H<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2,<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>) <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D4" target="_self" title="RECDEF_2:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span></span>)</span> <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E14:20_1"><i><font color="Green" title="E25">A10</font></i></a></i>
;<br/>

<a NAME="E11:20_1_9"/><i><font color="Green" title="E36">A27</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="recdef_2.html#K6" title="RECDEF_2:func.6">`3_4</a> 
 
<b>by </b><i><a class="txt" href="recdef_2.html#E17:20_1"><i><font color="Green" title="E27">A17</font></i></a>, <a class="txt" href="recdef_2.html#E1:20_1_9"><i><font color="Green" title="E33">A24</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_9_6"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1_9_6"/>
( ( <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 ) or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_9_6_1"><b>suppose </b></a><a NAME="E1:20_1_9_6_1"/>
( <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 ;<br/><div class="add"><a NAME="E2:20_1_9_6_1"/><b>then </b><i><font color="Green" title="E37">A28</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></i>;<br/><b>hence </b><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span></span>)</span> <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E17:20_1"><i><font color="Green" title="E27">A17</font></i></a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2,<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>) <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E14:20_1"><i><font color="Green" title="E25">A10</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span></span>)</span> <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D4" target="_self" title="RECDEF_2:def.4">Def4</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2,<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>) <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E14:20_1"><i><font color="Green" title="E25">A10</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K6" title="RECDEF_2:func.6">`3_4</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D5" target="_self" title="RECDEF_2:def.5">Def5</a></i>
<br/>.= 
<font color="Maroon">F<sub>4</sub></font>(1,<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()))
<b>by </b><i><a class="txt" href="recdef_2.html#E13:20_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="recdef_2.html#D6" target="_self" title="RECDEF_2:def.6">Def6</a></i>
<br/>.= 
<font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span>)
<b>by </b><i><a class="txt" href="recdef_2.html#E10:20_1"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="recdef_2.html#E20:20_1"><i><font color="Green" title="E29">A19</font></i></a>, <a class="txt" href="recdef_2.html#E21:20_1"><i><font color="Green" title="E30">A20</font></i></a>, <a class="txt" href="recdef_2.html#E1:20_1_9"><i><font color="Green" title="E33">A24</font></i></a>, <a class="txt" href="recdef_2.html#E2:20_1_9_6_1"><i><font color="Green" title="E37">A28</font></i></a>, <a class="ref" href="recdef_2.html#D7" target="_self" title="RECDEF_2:def.7">Def7</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_9_6_2"><b>suppose </b></a><a NAME="E1:20_1_9_6_2"/><i><font color="Green" title="E37">A29</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:20_1_9_6_2"/><b>then </b><i><font color="Green" title="E38">A30</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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">F<sub>4</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>())
 <b>by </b><i><a class="txt" href="recdef_2.html#E11:20_1"><i><font color="Green" title="E22">A7</font></i></a>, <a class="txt" href="recdef_2.html#E11:20_1_9"><i><font color="Green" title="E36">A27</font></i></a>, <a class="ref" href="recdef_2.html#D6" target="_self" title="RECDEF_2:def.6">Def6</a></i>;<br/><a NAME="E3:20_1_9_6_2"/><i><font color="Green" title="E39">A31</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(1,<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>4</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()))
 <b>by </b><i><a class="txt" href="recdef_2.html#E11:20_1"><i><font color="Green" title="E22">A7</font></i></a>, <a class="txt" href="recdef_2.html#E1:20_1_9"><i><font color="Green" title="E33">A24</font></i></a>, <a class="txt" href="recdef_2.html#E1:20_1_9_6_2"><i><font color="Green" title="E37">A29</font></i></a>, <a class="ref" href="recdef_2.html#D7" target="_self" title="RECDEF_2:def.7">Def7</a></i>;<br/><b>thus </b><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span></span>)</span> <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E17:20_1"><i><font color="Green" title="E27">A17</font></i></a>, <a class="txt" href="recdef_2.html#E1:20_1_9_6_2"><i><font color="Green" title="E37">A29</font></i></a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2,<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>) <a href="recdef_2.html#K4" title="RECDEF_2:func.4">`1_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E14:20_1"><i><font color="Green" title="E25">A10</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span></span>)</span> <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D4" target="_self" title="RECDEF_2:def.4">Def4</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2,<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>) <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E14:20_1"><i><font color="Green" title="E25">A10</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span></span>)</span> <a href="recdef_2.html#K6" title="RECDEF_2:func.6">`3_4</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D5" target="_self" title="RECDEF_2:def.5">Def5</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2,<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>) <a href="recdef_2.html#K6" title="RECDEF_2:func.6">`3_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E14:20_1"><i><font color="Green" title="E25">A10</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K7" title="RECDEF_2:func.7">`4_4</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D6" target="_self" title="RECDEF_2:def.6">Def6</a></i>
<br/>.= 
<font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span>)
<b>by </b><i><a class="txt" href="recdef_2.html#E13:20_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="recdef_2.html#E21:20_1"><i><font color="Green" title="E30">A20</font></i></a>, <a class="txt" href="recdef_2.html#E1:20_1_9_6_2"><i><font color="Green" title="E37">A29</font></i></a>, <a class="txt" href="recdef_2.html#E2:20_1_9_6_2"><i><font color="Green" title="E38">A30</font></i></a>, <a class="txt" href="recdef_2.html#E3:20_1_9_6_2"><i><font color="Green" title="E39">A31</font></i></a>, <a class="ref" href="recdef_2.html#D7" target="_self" title="RECDEF_2:def.7">Def7</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_9_6_3"><b>suppose </b></a><a NAME="E1:20_1_9_6_3"/><i><font color="Green" title="E37">A32</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font>
 ;<br/><div class="add"><a NAME="E2:20_1_9_6_3"/><b>then </b><i><font color="Green" title="E38">A33</font></i>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:20_1_9_6_3"/><b>then </b>
2 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 <b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E4:20_1_9_6_3"/><b>then </b><i><font color="Green" title="E39">A34</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2
 <b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E5:20_1_9_6_3"/>
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="recdef_2.html#E1:20_1_9_6_3"><i><font color="Green" title="E37">A32</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E6:20_1_9_6_3"/><b>then </b><i><font color="Green" title="E40">A35</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E7:20_1_9_6_3"/><i><font color="Green" title="E41">A36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 <b>by </b><i><a class="txt" href="recdef_2.html#E2:20_1_9_6_3"><i><font color="Green" title="E38">A33</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><a NAME="E8:20_1_9_6_3"/><i><font color="Green" title="E42">A37</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2
 ;<br/><b>thus </b><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E3:20_1_9"><i><font color="Green" title="E35">A26</font></i></a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2,<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>) <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E14:20_1"><i><font color="Green" title="E25">A10</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span></span>)</span> <a href="recdef_2.html#K6" title="RECDEF_2:func.6">`3_4</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D5" target="_self" title="RECDEF_2:def.5">Def5</a>, <a class="txt" href="recdef_2.html#E6:20_1_9_6_3"><i><font color="Green" title="E40">A35</font></i></a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="recdef_2.html#K6" title="RECDEF_2:func.6">`3_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E14:20_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="recdef_2.html#E6:20_1_9_6_3"><i><font color="Green" title="E40">A35</font></i></a>, <a class="txt" href="recdef_2.html#E8:20_1_9_6_3"><i><font color="Green" title="E42">A37</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span></span>)</span> <a href="recdef_2.html#K7" title="RECDEF_2:func.7">`4_4</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D6" target="_self" title="RECDEF_2:def.6">Def6</a>, <a class="txt" href="recdef_2.html#E4:20_1_9_6_3"><i><font color="Green" title="E39">A34</font></i></a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2,<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>) <a href="recdef_2.html#K7" title="RECDEF_2:func.7">`4_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E14:20_1"><i><font color="Green" title="E25">A10</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="recdef_2.html#K5" title="RECDEF_2:func.5">`2_4</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="recdef_2.html#K6" title="RECDEF_2:func.6">`3_4</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="recdef_2.html#K7" title="RECDEF_2:func.7">`4_4</a> </span>)</span>)
<b>by </b><i><a class="txt" href="recdef_2.html#E7:20_1_9_6_3"><i><font color="Green" title="E41">A36</font></i></a>, <a class="ref" href="recdef_2.html#D7" target="_self" title="RECDEF_2:def.7">Def7</a></i>
<br/>.= 
<font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span>)
<b>by </b><i><a class="txt" href="recdef_2.html#E17:20_1"><i><font color="Green" title="E27">A17</font></i></a>, <a class="txt" href="recdef_2.html#E1:20_1_9"><i><font color="Green" title="E33">A24</font></i></a>, <a class="txt" href="recdef_2.html#E11:20_1_9"><i><font color="Green" title="E36">A27</font></i></a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E24:20_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="recdef_2.html#E22:20_1"><i><font color="Green" title="E31">A21</font></i></a>, <a class="txt" href="recdef_2.html#E23:20_1"><i><font color="Green" title="E32">A23</font></i></a>);<br/></i>
<b>hence </b><a NAME="E25:20_1"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<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></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span>)
 ;<br/>


</div>
