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

<b>set </b><font color="Maroon">c<sub>1</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">c<sub>2</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>set </b><font color="Maroon">c<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>())));<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="recdef_2.html#K5" target="_self">`2_4</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="recdef_2.html#K6" target="_self">`3_4</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="recdef_2.html#K7" target="_self">`4_4</a> </span>)</span>,<font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="recdef_2.html#K5" target="_self">`2_4</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="recdef_2.html#K6" target="_self">`3_4</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="recdef_2.html#K7" target="_self">`4_4</a> </span>)</span>)</span><a href="mcart_1.html#K4">]</a></span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 2,<font color="Maroon">a<sub>2</sub></font>);<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E2:20_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E3:20_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K4">[</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">]</a></span>
 <b>and </b><br/><a NAME="E4:20_1"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="recdef_1.html#S3">RECDEF_1:sch 3</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ] means  <a href="funct_7.html#K1">In</a> <font color="Maroon">a<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1"><a href="mcart_1.html#K4">[</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">]</a></span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1"><a href="mcart_1.html#K4">[</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">]</a></span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_7.html#K1">In</a> <font color="Maroon">a<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a> </span>)</span> <a href="binarith.html#K5">-'</a> 2</span>)</span>;<br/>
<a NAME="E5:20_1"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies  not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies  not <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) &amp; ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies  not <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:20_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

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


</div><b>end;</b></div>
<a NAME="E6:20_1"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  or <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] or <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:20_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<a NAME="E2:20_1_2"/><b>then </b>
 <a href="funct_7.html#K1">In</a> <font color="Maroon">c<sub>5</sub></font>,<a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="funct_7.html#D1">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">c<sub>5</sub></font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>5</sub></font>] or <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>5</sub></font>] )
 <b>by </b><i><a class="ref" href="cqc_the1.html#T2">CQC_THE1:2</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:20_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E9:20_1"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) &amp; ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) &amp; ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="recdef_2.html#S1" target="_self">RECDEF_2:sch 1</a>(<a class="txt" href="recdef_2.html#E5:20_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="recdef_2.html#E6:20_1"><i><font color="Green">E19</font></i></a>);<br/></i>
<a NAME="E10:20_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K4">[</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">]</a></span>
 
<b>by </b><i><a class="txt" href="recdef_2.html#E9:20_1"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E11:20_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K4">[</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">]</a></span>
 
<b>by </b><i><a class="txt" href="recdef_2.html#E9:20_1"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E12:20_1"/><i><font color="Green">E23</font></i>: 
 <a href="funct_7.html#K1">In</a> 2,<a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="funct_7.html#D1">FUNCT_7:def 1</a></i>;<br/>
<b>then </b><i><font color="Green">E24</font></i>: <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</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">E20</font></i></a></i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K4">[</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">]</a></span>
<b>by </b><i><a class="txt" href="recdef_2.html#E3:20_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="recdef_2.html#E12:20_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
;<br/>
<a NAME="E14:20_1"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 3</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 2,<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</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">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:20_1_4"/><i><font color="Green">E26</font></i>: 
(  <a href="funct_7.html#K1">In</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 2 &amp;  <a href="funct_7.html#K1">In</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 3</span>)</span>,<a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 3 )
 
<b>by </b><i><a class="ref" href="funct_7.html#D1">FUNCT_7:def 1</a></i>;<br/>
<a NAME="E2:20_1_4"/><i><font color="Green">E27</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E3:20_1_4"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>then </b><i><font color="Green">E28</font></i>: <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> = 
<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 2)
<b>by </b><i><a class="txt" href="recdef_2.html#E9:20_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="recdef_2.html#E1:20_1_4"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="recdef_2.html#E1:20_1_4"><i><font color="Green">E26</font></i></a>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
;<br/>
<a NAME="E5:20_1_4"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 3</span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">3 <a href="xcmplx_0.html#K6">-</a> 2</span>)</span>
 
;<br/>
<a NAME="E6:20_1_4"/><i><font color="Green">E30</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E7:20_1_4"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 3
 
<b>by </b><i><a class="txt" href="recdef_2.html#E2:20_1_4"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 3</span>)</span> = 
<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 3)
<b>by </b><i><a class="txt" href="recdef_2.html#E9:20_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="recdef_2.html#E1:20_1_4"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="recdef_2.html#E1:20_1_4"><i><font color="Green">E26</font></i></a>, <a class="txt" href="recdef_2.html#E5:20_1_4"><i><font color="Green">E29</font></i></a>, <a class="txt" href="recdef_2.html#E6:20_1_4"><i><font color="Green">E30</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 2,<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span>)
<b>by </b><i><a class="txt" href="recdef_2.html#E4:20_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="recdef_2.html#E4:20_1_4"><i><font color="Green">E28</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">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="recdef_2.html#K4" target="_self">`1_4</a> ;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E16:20_1"/><i><font color="Green">E26</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E17:20_1"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>6</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S4">FUNCT_1:sch 4</a>();<br/></i>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>thus </b><a NAME="E18:20_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="recdef_2.html#E16:20_1"><i><font color="Green">E26</font></i></a></i>;<br/>

<b>thus </b><i><font color="Green">E28</font></i>: <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> 0 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 0</span>)</span> <a href="recdef_2.html#K4" target="_self">`1_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E17:20_1"><i><font color="Green">E27</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">E21</font></i></a>, <a class="ref" href="recdef_2.html#D4" target="_self">Def4</a></i>
;<br/>

<b>thus </b><i><font color="Green">E29</font></i>: <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="recdef_2.html#K4" target="_self">`1_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E17:20_1"><i><font color="Green">E27</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">E22</font></i></a>, <a class="ref" href="recdef_2.html#D4" target="_self">Def4</a></i>
;<br/>

<b>thus </b><i><font color="Green">E30</font></i>: <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> 2 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 2</span>)</span> <a href="recdef_2.html#K4" target="_self">`1_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E17:20_1"><i><font color="Green">E27</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">E24</font></i></a>, <a class="ref" href="recdef_2.html#D4" target="_self">Def4</a></i>
;<br/>

<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 3</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K5" target="_self">`2_4</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="recdef_2.html#K5" target="_self">`2_4</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K4" target="_self">`1_4</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="recdef_2.html#K6" target="_self">`3_4</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K5" target="_self">`2_4</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="recdef_2.html#K7" target="_self">`4_4</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K6" target="_self">`3_4</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K5" target="_self">`2_4</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K4" target="_self">`1_4</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K6" target="_self">`3_4</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K5" target="_self">`2_4</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K7" target="_self">`4_4</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K6" target="_self">`3_4</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K5" target="_self">`2_4</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span> <a href="recdef_2.html#K4" target="_self">`1_4</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 3</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">a<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>) );<br/>
<a NAME="E22:20_1"/><i><font color="Green">E31</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">E32</font></i>: <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 3</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span> <a href="recdef_2.html#K4" target="_self">`1_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E17:20_1"><i><font color="Green">E27</font></i></a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(0 <a href="nat_1.html#K1">+</a> 2,<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 2</span>)</span>) <a href="recdef_2.html#K4" target="_self">`1_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E14:20_1"><i><font color="Green">E25</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K5" target="_self">`2_4</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D4" target="_self">Def4</a></i>
;<br/>

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

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 0</span>)</span> <a href="recdef_2.html#K6" target="_self">`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">E21</font></i></a>, <a class="ref" href="recdef_2.html#D6" target="_self">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K5" target="_self">`2_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E11:20_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="recdef_2.html#D5" target="_self">Def5</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 0</span>)</span> <a href="recdef_2.html#K7" target="_self">`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">E21</font></i></a>, <a class="ref" href="recdef_2.html#D7" target="_self">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K6" target="_self">`3_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E11:20_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="recdef_2.html#D6" target="_self">Def6</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K5" target="_self">`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">E22</font></i></a>, <a class="ref" href="recdef_2.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K4" target="_self">`1_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E13:20_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="recdef_2.html#D4" target="_self">Def4</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K6" target="_self">`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">E22</font></i></a>, <a class="ref" href="recdef_2.html#D6" target="_self">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K5" target="_self">`2_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E13:20_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="recdef_2.html#D5" target="_self">Def5</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K7" target="_self">`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">E22</font></i></a>, <a class="ref" href="recdef_2.html#D7" target="_self">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K6" target="_self">`3_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E13:20_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="recdef_2.html#D6" target="_self">Def6</a></i>
;<br/>

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

<b>thus </b><a NAME="E9:20_1_8"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 3</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(0,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>)
 <b>by </b><i><a class="txt" href="recdef_2.html#E13:20_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="recdef_2.html#E19:20_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="recdef_2.html#E20:20_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="recdef_2.html#E21:20_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="recdef_2.html#E1:20_1_8"><i><font color="Green">E32</font></i></a>, <a class="ref" href="recdef_2.html#D5" target="_self">Def5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:20_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</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">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:20_1_9"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>7</sub></font>]
 ;<br/>

<a NAME="E2:20_1_9"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 2</span>)</span>
 
;<br/>
<b>thus </b><i><font color="Green">E35</font></i>: <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span> <a href="recdef_2.html#K4" target="_self">`1_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E17:20_1"><i><font color="Green">E27</font></i></a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2,<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>) <a href="recdef_2.html#K4" target="_self">`1_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E14:20_1"><i><font color="Green">E25</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K5" target="_self">`2_4</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D4" target="_self">Def4</a></i>
;<br/>

<b>thus </b><a NAME="E4:20_1_9"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K4" target="_self">`1_4</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K5" target="_self">`2_4</a> 
 <b>by </b><i><a class="txt" href="recdef_2.html#E1:20_1_9"><i><font color="Green">E33</font></i></a></i>;<br/>

<b>thus </b><a NAME="E5:20_1_9"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K6" target="_self">`3_4</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K5" target="_self">`2_4</a> 
 <b>by </b><i><a class="txt" href="recdef_2.html#E1:20_1_9"><i><font color="Green">E33</font></i></a></i>;<br/>

<b>thus </b><a NAME="E6:20_1_9"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K7" target="_self">`4_4</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K6" target="_self">`3_4</a> 
 <b>by </b><i><a class="txt" href="recdef_2.html#E1:20_1_9"><i><font color="Green">E33</font></i></a></i>;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K5" target="_self">`2_4</a>  = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 2,<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span>) <a href="recdef_2.html#K4" target="_self">`1_4</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D4" target="_self">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K4" target="_self">`1_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E14:20_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="recdef_2.html#E2:20_1_9"><i><font color="Green">E34</font></i></a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K6" target="_self">`3_4</a>  = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 2,<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span>) <a href="recdef_2.html#K5" target="_self">`2_4</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K5" target="_self">`2_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E14:20_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="recdef_2.html#E2:20_1_9"><i><font color="Green">E34</font></i></a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K7" target="_self">`4_4</a>  = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 2,<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span>) <a href="recdef_2.html#K6" target="_self">`3_4</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D6" target="_self">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K6" target="_self">`3_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E14:20_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="recdef_2.html#E2:20_1_9"><i><font color="Green">E34</font></i></a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K5" target="_self">`2_4</a>  = 
<font color="Maroon">H<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2,<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>) <a href="recdef_2.html#K4" target="_self">`1_4</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D4" target="_self">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span> <a href="recdef_2.html#K4" target="_self">`1_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E14:20_1"><i><font color="Green">E25</font></i></a></i>
;<br/>

<a NAME="E11:20_1_9"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="recdef_2.html#K6" target="_self">`3_4</a> 
 
<b>by </b><i><a class="txt" href="recdef_2.html#E17:20_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="recdef_2.html#E1:20_1_9"><i><font color="Green">E33</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">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 ) or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 or 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></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">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 ;<br/><div class="add"><a NAME="E2:20_1_9_6_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="cqc_the1.html#T2">CQC_THE1:2</a></i>;<br/><b>hence </b><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span> <a href="recdef_2.html#K4" target="_self">`1_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E17:20_1"><i><font color="Green">E27</font></i></a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(1 <a href="nat_1.html#K1">+</a> 2,<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 2</span>)</span>) <a href="recdef_2.html#K4" target="_self">`1_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E14:20_1"><i><font color="Green">E25</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span> <a href="recdef_2.html#K5" target="_self">`2_4</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D4" target="_self">Def4</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(0 <a href="nat_1.html#K1">+</a> 2,<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 2</span>)</span>) <a href="recdef_2.html#K5" target="_self">`2_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E14:20_1"><i><font color="Green">E25</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K6" target="_self">`3_4</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D5" target="_self">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">E24</font></i></a>, <a class="ref" href="recdef_2.html#D6" target="_self">Def6</a></i>
<br/>.= 
<font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>)
<b>by </b><i><a class="txt" href="recdef_2.html#E10:20_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="recdef_2.html#E20:20_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="recdef_2.html#E21:20_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="recdef_2.html#E1:20_1_9"><i><font color="Green">E33</font></i></a>, <a class="txt" href="recdef_2.html#E2:20_1_9_6_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="recdef_2.html#D7" target="_self">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">E37</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:20_1_9_6_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</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">E22</font></i></a>, <a class="txt" href="recdef_2.html#E11:20_1_9"><i><font color="Green">E36</font></i></a>, <a class="ref" href="recdef_2.html#D6" target="_self">Def6</a></i>;<br/><a NAME="E3:20_1_9_6_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</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">E22</font></i></a>, <a class="txt" href="recdef_2.html#E1:20_1_9"><i><font color="Green">E33</font></i></a>, <a class="txt" href="recdef_2.html#E1:20_1_9_6_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="recdef_2.html#D7" target="_self">Def7</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span> <a href="recdef_2.html#K4" target="_self">`1_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E17:20_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="recdef_2.html#E1:20_1_9_6_2"><i><font color="Green">E37</font></i></a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(2 <a href="nat_1.html#K1">+</a> 2,<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 2</span>)</span>) <a href="recdef_2.html#K4" target="_self">`1_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E14:20_1"><i><font color="Green">E25</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span> <a href="recdef_2.html#K5" target="_self">`2_4</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D4" target="_self">Def4</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(1 <a href="nat_1.html#K1">+</a> 2,<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 2</span>)</span>) <a href="recdef_2.html#K5" target="_self">`2_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E14:20_1"><i><font color="Green">E25</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span> <a href="recdef_2.html#K6" target="_self">`3_4</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D5" target="_self">Def5</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(0 <a href="nat_1.html#K1">+</a> 2,<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 2</span>)</span>) <a href="recdef_2.html#K6" target="_self">`3_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E14:20_1"><i><font color="Green">E25</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K7" target="_self">`4_4</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D6" target="_self">Def6</a></i>
<br/>.= 
<font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>)
<b>by </b><i><a class="txt" href="recdef_2.html#E13:20_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="recdef_2.html#E21:20_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="recdef_2.html#E1:20_1_9_6_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="recdef_2.html#E2:20_1_9_6_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="recdef_2.html#E3:20_1_9_6_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="recdef_2.html#D7" target="_self">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">E37</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><a NAME="E2:20_1_9_6_3"/><b>then </b><i><font color="Green">E38</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:20_1_9_6_3"/><b>then </b>
2 <a href="xcmplx_0.html#K6">-</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K6">-</a> 2
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E4:20_1_9_6_3"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K6">-</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 2
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E5:20_1_9_6_3"/>
1 <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="recdef_2.html#E1:20_1_9_6_3"><i><font color="Green">E37</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E6:20_1_9_6_3"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E7:20_1_9_6_3"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="recdef_2.html#E2:20_1_9_6_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E8:20_1_9_6_3"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 ;<br/><b>thus </b><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="recdef_2.html#K5" target="_self">`2_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E3:20_1_9"><i><font color="Green">E35</font></i></a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 2,<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span>) <a href="recdef_2.html#K5" target="_self">`2_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E14:20_1"><i><font color="Green">E25</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="recdef_2.html#K6" target="_self">`3_4</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span> <a href="recdef_2.html#K6" target="_self">`3_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E6:20_1_9_6_3"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1,<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>) <a href="recdef_2.html#K6" target="_self">`3_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E14:20_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="recdef_2.html#E6:20_1_9_6_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="recdef_2.html#E8:20_1_9_6_3"><i><font color="Green">E42</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="recdef_2.html#K7" target="_self">`4_4</a> 
<b>by </b><i><a class="ref" href="recdef_2.html#D6" target="_self">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span> <a href="recdef_2.html#K7" target="_self">`4_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E4:20_1_9_6_3"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 2,<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>) <a href="recdef_2.html#K7" target="_self">`4_4</a> 
<b>by </b><i><a class="txt" href="recdef_2.html#E14:20_1"><i><font color="Green">E25</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="recdef_2.html#K5" target="_self">`2_4</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="recdef_2.html#K6" target="_self">`3_4</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="recdef_2.html#K7" target="_self">`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">E41</font></i></a>, <a class="ref" href="recdef_2.html#D7" target="_self">Def7</a></i>
<br/>.= 
<font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>)
<b>by </b><i><a class="txt" href="recdef_2.html#E17:20_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="recdef_2.html#E1:20_1_9"><i><font color="Green">E33</font></i></a>, <a class="txt" href="recdef_2.html#E11:20_1_9"><i><font color="Green">E36</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">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="recdef_2.html#E22:20_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="recdef_2.html#E23:20_1"><i><font color="Green">E32</font></i></a>);<br/></i>
<b>hence </b><a NAME="E25:20_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 3</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>)
 ;<br/>


</div>
