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

<b>set </b><font color="Maroon" title="c1">r03</font> = <font color="Maroon">F<sub>3</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>());<br/>
<b>set </b><font color="Maroon" title="c2">r13</font> = <font color="Maroon">F<sub>3</sub></font>(1,<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</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#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2">(<span class="default">$2 <a href="recdef_2.html#K2" title="RECDEF_2:func.2">`2_3</a> </span>)</span>,<span class="p2">(<span class="default">$2 <a href="recdef_2.html#K3" title="RECDEF_2:func.3">`3_3</a> </span>)</span>,<font color="Maroon">F<sub>3</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#K2" title="RECDEF_2:func.2">`2_3</a> </span>)</span>,<span class="p1">(<span class="default">$2 <a href="recdef_2.html#K3" title="RECDEF_2:func.3">`3_3</a> </span>)</span>)</span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</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> 1,$2);<br/>
<b>consider </b><font color="Maroon" title="c3">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E2:16_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">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="E3:16_1"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c3">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#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</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>(1,<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()))</span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 <b>and </b><br/><a NAME="E4:16_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="c3">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="c3">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>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#K3" title="MCART_1:func.3">[</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>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>())</span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>;<br/>
<b>deffunc </b><font color="Maroon">H<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>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c4">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E6:16_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">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="E7:16_1"/><i><font color="Green" title="E18">A3</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 <br/> ( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] implies <font color="Maroon" title="c4">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; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] implies <font color="Maroon" title="c4">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>) ) )
 <b>from </b><i><a class="ref" href="partfun1.html#S4" title="PARTFUN1:sch.4">PARTFUN1:sch 4</a>();<br/></i>
<a NAME="E8:16_1"/><i><font color="Green" title="E19">A4</font></i>: 
<font color="Maroon" title="c4">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#K3" title="MCART_1:func.3">[</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>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>())</span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 
<b>by </b><i><a class="txt" href="recdef_2.html#E7:16_1"><i><font color="Green" title="E18">A3</font></i></a></i>;<br/>
<i><font color="Green" title="E20">A5</font></i>: <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<font color="Maroon">H<sub>4</sub></font>(1)
<b>by </b><i><a class="txt" href="recdef_2.html#E7:16_1"><i><font color="Green" title="E18">A3</font></i></a></i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</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>(1,<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()))</span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
<b>by </b><i><a class="txt" href="recdef_2.html#E3:16_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>
;<br/>
<a NAME="E10:16_1"/><i><font color="Green" title="E21">A6</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">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> <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> 1,<font color="Maroon" title="c4">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> 1</span>)</span>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">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/>

<i><font color="Green" title="E22">A7</font></i>: <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1)
<b>by </b><i><a class="txt" href="recdef_2.html#E7:16_1"><i><font color="Green" title="E18">A3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font>
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>
;<br/>
<a NAME="E2:16_1_2"/><i><font color="Green" title="E23">A8</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <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="c5">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 
;<br/>
<a NAME="E3:16_1_2"/><i><font color="Green" title="E24">A9</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">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/>
<b>thus </b><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> = 
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c5">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#E7:16_1"><i><font color="Green" title="E18">A3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">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#E2:16_1_2"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="recdef_2.html#E3:16_1_2"><i><font color="Green" title="E24">A9</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="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">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#E4:16_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="recdef_2.html#E1:16_1_2"><i><font color="Green" title="E22">A7</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<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>  = <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="recdef_2.html#K1" title="RECDEF_2:func.1">`1_3</a> ;<br/>
<b>consider </b><font color="Maroon" title="c5">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E12:16_1"/><i><font color="Green" title="E22">A10</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">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="E13:16_1"/><i><font color="Green" title="E23">A11</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" title="c5">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>5</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i>
<b>take </b>
<font color="Maroon" title="c5">f</font>
;<br/>

<b>thus </b><a NAME="E14:16_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">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#E12:16_1"><i><font color="Green" title="E22">A10</font></i></a></i>;<br/>

<b>thus </b><i><font color="Green" title="E24">A12</font></i>: <font color="Maroon" title="c5">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="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0</span>)</span> <a href="recdef_2.html#K1" title="RECDEF_2:func.1">`1_3</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E13:16_1"><i><font color="Green" title="E23">A11</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>1</sub></font>()
<b>by </b><i><a class="txt" href="recdef_2.html#E8:16_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="recdef_2.html#D1" target="_self" title="RECDEF_2:def.1">Def1</a></i>
;<br/>

<b>thus </b><i><font color="Green" title="E25">A13</font></i>: <font color="Maroon" title="c5">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="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="recdef_2.html#K1" title="RECDEF_2:func.1">`1_3</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E13:16_1"><i><font color="Green" title="E23">A11</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>2</sub></font>()
<b>by </b><i><a class="txt" href="recdef_2.html#E9:16_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="recdef_2.html#D1" target="_self" title="RECDEF_2:def.1">Def1</a></i>
;<br/>

<b>defpred </b><font color="Maroon">S<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> ] <b>means </b>( <font color="Maroon" title="c5">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> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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#K2" title="RECDEF_2:func.2">`2_3</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">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#K1" title="RECDEF_2:func.1">`1_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="recdef_2.html#K2" title="RECDEF_2:func.2">`2_3</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">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#K1" title="RECDEF_2:func.1">`1_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="recdef_2.html#K3" title="RECDEF_2:func.3">`3_3</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">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#K1" title="RECDEF_2:func.1">`1_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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#K2" title="RECDEF_2:func.2">`2_3</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">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#K2" title="RECDEF_2:func.2">`2_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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#K3" title="RECDEF_2:func.3">`3_3</a>  &amp; <font color="Maroon" title="c5">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> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>($1,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c5">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>) );<br/>
<a NAME="E17:16_1"/><i><font color="Green" title="E26">A14</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_5"><b>proof </b></a><div class="add">

<b>thus </b><i><font color="Green" title="E27">A15</font></i>: <font color="Maroon" title="c5">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> 2</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">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#K1" title="RECDEF_2:func.1">`1_3</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E13:16_1"><i><font color="Green" title="E23">A11</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> 1,<font color="Maroon" title="c4">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> 1</span>)</span>) <a href="recdef_2.html#K1" title="RECDEF_2:func.1">`1_3</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E10:16_1"><i><font color="Green" title="E21">A6</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">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#K2" title="RECDEF_2:func.2">`2_3</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D1" target="_self" title="RECDEF_2:def.1">Def1</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">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#K1" title="RECDEF_2:func.1">`1_3</a>  = 
<font color="Maroon">F<sub>2</sub></font>()
<b>by </b><i><a class="txt" href="recdef_2.html#E9:16_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="recdef_2.html#D1" target="_self" title="RECDEF_2:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0</span>)</span> <a href="recdef_2.html#K2" title="RECDEF_2:func.2">`2_3</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E8:16_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="recdef_2.html#D2" target="_self" title="RECDEF_2:def.2">Def2</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">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#K1" title="RECDEF_2:func.1">`1_3</a>  = 
<font color="Maroon">H<sub>1</sub></font>(0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c4">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> 1</span>)</span>) <a href="recdef_2.html#K1" title="RECDEF_2:func.1">`1_3</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E10:16_1"><i><font color="Green" title="E21">A6</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="recdef_2.html#K2" title="RECDEF_2:func.2">`2_3</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D1" target="_self" title="RECDEF_2:def.1">Def1</a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>())
<b>by </b><i><a class="txt" href="recdef_2.html#E9:16_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="recdef_2.html#D2" target="_self" title="RECDEF_2:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0</span>)</span> <a href="recdef_2.html#K3" title="RECDEF_2:func.3">`3_3</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E8:16_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="recdef_2.html#D3" target="_self" title="RECDEF_2:def.3">Def3</a></i>
;<br/>

<b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">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#K1" title="RECDEF_2:func.1">`1_3</a>  = 
<font color="Maroon">F<sub>3</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>())
<b>by </b><i><a class="txt" href="recdef_2.html#E8:16_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="recdef_2.html#D3" target="_self" title="RECDEF_2:def.3">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">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#K2" title="RECDEF_2:func.2">`2_3</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E9:16_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="recdef_2.html#D2" target="_self" title="RECDEF_2:def.2">Def2</a></i>
;<br/><br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">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#K2" title="RECDEF_2:func.2">`2_3</a>  = 
<font color="Maroon">H<sub>1</sub></font>(0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1,<font color="Maroon" title="c4">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> 1</span>)</span>) <a href="recdef_2.html#K2" title="RECDEF_2:func.2">`2_3</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E10:16_1"><i><font color="Green" title="E21">A6</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">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#K3" title="RECDEF_2:func.3">`3_3</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D2" target="_self" title="RECDEF_2:def.2">Def2</a></i>
;<br/>

<b>thus </b><a NAME="E6:16_1_5"/>
<font color="Maroon" title="c5">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> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(0,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c5">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>)
 <b>by </b><i><a class="txt" href="recdef_2.html#E9:16_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="recdef_2.html#E15:16_1"><i><font color="Green" title="E24">A12</font></i></a>, <a class="txt" href="recdef_2.html#E16:16_1"><i><font color="Green" title="E25">A13</font></i></a>, <a class="txt" href="recdef_2.html#E1:16_1_5"><i><font color="Green" title="E27">A15</font></i></a>, <a class="ref" href="recdef_2.html#D2" target="_self" title="RECDEF_2:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:16_1"/><i><font color="Green" title="E27">A16</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>2</sub></font>[<font color="Olive" title="b1">x</font>] holds <br/><font color="Maroon">S<sub>2</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="16_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">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="E1:16_1_6"/><i><font color="Green" title="E28">A17</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">x</font>]
 ;<br/>

<b>thus </b><i><font color="Green" title="E29">A18</font></i>: <font color="Maroon" title="c5">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="c6">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 class="p1">(<span class="default"><font color="Maroon" title="c4">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="c6">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#K1" title="RECDEF_2:func.1">`1_3</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E13:16_1"><i><font color="Green" title="E23">A11</font></i></a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c6">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,<font color="Maroon" title="c4">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="c6">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="recdef_2.html#K1" title="RECDEF_2:func.1">`1_3</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E10:16_1"><i><font color="Green" title="E21">A6</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">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="c6">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#K2" title="RECDEF_2:func.2">`2_3</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D1" target="_self" title="RECDEF_2:def.1">Def1</a></i>
;<br/>

<b>thus </b><a NAME="E3:16_1_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">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="c6">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#K1" title="RECDEF_2:func.1">`1_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K2" title="RECDEF_2:func.2">`2_3</a> 
 <b>by </b><i><a class="txt" href="recdef_2.html#E1:16_1_6"><i><font color="Green" title="E28">A17</font></i></a></i>;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">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="c6">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#K1" title="RECDEF_2:func.1">`1_3</a>  = 
<font color="Maroon">H<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c6">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,<font color="Maroon" title="c4">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="c6">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="recdef_2.html#K1" title="RECDEF_2:func.1">`1_3</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E10:16_1"><i><font color="Green" title="E21">A6</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K3" title="RECDEF_2:func.3">`3_3</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E1:16_1_6"><i><font color="Green" title="E28">A17</font></i></a>, <a class="ref" href="recdef_2.html#D1" target="_self" title="RECDEF_2:def.1">Def1</a></i>
;<br/>

<b>hence </b><a NAME="E5:16_1_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">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="c6">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#K1" title="RECDEF_2:func.1">`1_3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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="c6">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#K2" title="RECDEF_2:func.2">`2_3</a> 
 <b>by </b><i><a class="txt" href="recdef_2.html#E1:16_1_6"><i><font color="Green" title="E28">A17</font></i></a></i>;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">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="c6">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#K2" title="RECDEF_2:func.2">`2_3</a>  = 
<font color="Maroon">H<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c6">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,<font color="Maroon" title="c4">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="c6">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="recdef_2.html#K2" title="RECDEF_2:func.2">`2_3</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E10:16_1"><i><font color="Green" title="E21">A6</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">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="c6">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#K3" title="RECDEF_2:func.3">`3_3</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D2" target="_self" title="RECDEF_2:def.2">Def2</a></i>
;<br/>

<a NAME="E7:16_1_6"/><i><font color="Green" title="E30">A19</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="recdef_2.html#K2" title="RECDEF_2:func.2">`2_3</a> 
 
<b>by </b><i><a class="txt" href="recdef_2.html#E13:16_1"><i><font color="Green" title="E23">A11</font></i></a>, <a class="txt" href="recdef_2.html#E1:16_1_6"><i><font color="Green" title="E28">A17</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_6_4"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_6_4"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_6_4_1"><b>suppose </b></a><a NAME="E1:16_1_6_4_1"/><i><font color="Green" title="E31">A20</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c5">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="c6">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 class="p1">(<span class="default"><font color="Maroon" title="c4">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#K1" title="RECDEF_2:func.1">`1_3</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E13:16_1"><i><font color="Green" title="E23">A11</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> 1,<font color="Maroon" title="c4">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> 1</span>)</span>) <a href="recdef_2.html#K1" title="RECDEF_2:func.1">`1_3</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E10:16_1"><i><font color="Green" title="E21">A6</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">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#K3" title="RECDEF_2:func.3">`3_3</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E1:16_1_6"><i><font color="Green" title="E28">A17</font></i></a>, <a class="txt" href="recdef_2.html#E1:16_1_6_4_1"><i><font color="Green" title="E31">A20</font></i></a>, <a class="ref" href="recdef_2.html#D1" target="_self" title="RECDEF_2:def.1">Def1</a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(1,<font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(0,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()))
<b>by </b><i><a class="txt" href="recdef_2.html#E9:16_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="recdef_2.html#D3" target="_self" title="RECDEF_2:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0</span>)</span> <a href="recdef_2.html#K3" title="RECDEF_2:func.3">`3_3</a> </span>)</span>)
<b>by </b><i><a class="txt" href="recdef_2.html#E8:16_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="recdef_2.html#D3" target="_self" title="RECDEF_2:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">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="c5">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="c6">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>)
<b>by </b><i><a class="txt" href="recdef_2.html#E8:16_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="recdef_2.html#E1:16_1_6"><i><font color="Green" title="E28">A17</font></i></a>, <a class="txt" href="recdef_2.html#E7:16_1_6"><i><font color="Green" title="E30">A19</font></i></a>, <a class="txt" href="recdef_2.html#E1:16_1_6_4_1"><i><font color="Green" title="E31">A20</font></i></a>, <a class="ref" href="recdef_2.html#D2" target="_self" title="RECDEF_2:def.2">Def2</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_6_4_2"><b>suppose </b></a><a NAME="E1:16_1_6_4_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:16_1_6_4_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/><a NAME="E3:16_1_6_4_2"/><b>then </b><i><font color="Green" title="E31">A21</font></i>: 
0 <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="c6">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="E4:16_1_6_4_2"/><b>then </b>
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="c6">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <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="E5:16_1_6_4_2"/><b>then </b><i><font color="Green" title="E32">A22</font></i>: 
<font color="Maroon" title="c6">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="c6">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="E6:16_1_6_4_2"/><i><font color="Green" title="E33">A23</font></i>: 
<font color="Maroon" title="c6">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="c6">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/><a NAME="E7:16_1_6_4_2"/><i><font color="Green" title="E34">A24</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="recdef_2.html#E3:16_1_6_4_2"><i><font color="Green" title="E31">A21</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><b>thus </b><font color="Maroon" title="c5">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="c6">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> = 
<font color="Maroon">H<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c6">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> 1,<font color="Maroon" title="c4">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="c6">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> 1</span>)</span>) <a href="recdef_2.html#K3" title="RECDEF_2:func.3">`3_3</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E2:16_1_6"><i><font color="Green" title="E29">A18</font></i></a>, <a class="txt" href="recdef_2.html#E10:16_1"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="recdef_2.html#E1:16_1_6"><i><font color="Green" title="E28">A17</font></i></a>, <a class="txt" href="recdef_2.html#E5:16_1_6_4_2"><i><font color="Green" title="E32">A22</font></i></a>, <a class="txt" href="recdef_2.html#E6:16_1_6_4_2"><i><font color="Green" title="E33">A23</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">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="c5">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="c6">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>)
<b>by </b><i><a class="txt" href="recdef_2.html#E1:16_1_6"><i><font color="Green" title="E28">A17</font></i></a>, <a class="txt" href="recdef_2.html#E7:16_1_6"><i><font color="Green" title="E30">A19</font></i></a>, <a class="txt" href="recdef_2.html#E7:16_1_6_4_2"><i><font color="Green" title="E34">A24</font></i></a>, <a class="ref" href="recdef_2.html#D3" target="_self" title="RECDEF_2:def.3">Def3</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E19:16_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>2</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#E17:16_1"><i><font color="Green" title="E26">A14</font></i></a>, <a class="txt" href="recdef_2.html#E18:16_1"><i><font color="Green" title="E27">A16</font></i></a>);<br/></i>
<b>hence </b><a NAME="E20:16_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="c5">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> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c5">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>)
 ;<br/>


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