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

<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">c</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>let </b><font color="Maroon" title="c3">P</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="comput_1.html#V7" title="COMPUT_1:attr.7">primitive-recursively_closed</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="comput_1.html#K3" title="COMPUT_1:func.3">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>;<br/>





<a NAME="E1:115"/><i><font color="Green" title="E79">A1</font></i>: 
<font color="Maroon" title="c3">P</font> is <a href="comput_1.html#V5" title="COMPUT_1:attr.5">composition_closed</a>
 
<b>by </b><i><a class="ref" href="comput_1.html#D17" target="_self" title="COMPUT_1:def.17">Def17</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><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>;<br/>
<a NAME="E2:115"/><i><font color="Green" title="E80">A2</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="ref" href="comput_1.html#D17" target="_self" title="COMPUT_1:def.17">Def17</a></i>;<br/>
<a NAME="E3:115"/><i><font color="Green" title="E81">A3</font></i>: 
for <font color="Olive" title="b1">i</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">i</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</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>set </b><font color="Maroon" title="c5">F</font> = <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c4">i</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">0consti</font> = <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c4">i</font> as    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K3" title="COMPUT_1:func.3">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="comput_1.html#T31" target="_self" title="COMPUT_1:th.31">Th31</a></i>;<br/>
<a NAME="E2:115_1"/>
<span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c6">0consti</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="comput_1.html#K3" title="COMPUT_1:func.3">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">F</font> = <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c4">i</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> as  <a href="comput_1.html#V4" title="COMPUT_1:attr.4">with_the_same_arity</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="comput_1.html#K3" title="COMPUT_1:func.3">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<b>assume </b><a NAME="E4:115_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 ;<br/>

<a NAME="E5:115_1"/><b>then </b><i><font color="Green" title="E82">A4</font></i>: 
( <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c4">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c4">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E6:115_1"/><i><font color="Green" title="E83">A5</font></i>: 
1 <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="ref" href="comput_1.html#D17" target="_self" title="COMPUT_1:def.17">Def17</a></i>;<br/>
<i><font color="Green" title="E84">A6</font></i>:  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <span class="p1">(<span class="default">1 <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> 1</span>)</span> = 
1
<b>by </b><i><a class="ref" href="comput_1.html#T38" target="_self" title="COMPUT_1:th.38">Th38</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c8">1succ1</font> = 1 <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> 1 as  <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>  <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K3" title="COMPUT_1:func.3">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="comput_1.html#T32" target="_self" title="COMPUT_1:th.32">Th32</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">h</font> be    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K3" title="COMPUT_1:func.3">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:115_1_2"/>
<font color="Maroon" title="c9">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F</font>
 ;<br/><a NAME="E2:115_1_2"/><b>then </b>
<font color="Maroon" title="c9">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E5:115_1"><i><font color="Green" title="E82">A4</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:115_1_2"/>
<font color="Maroon" title="c9">h</font> is <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>
 <b>by </b><i><a class="ref" href="comput_1.html#T32" target="_self" title="COMPUT_1:th.32">Th32</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c8">1succ1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c7">F</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> as  <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>  <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K3" title="COMPUT_1:func.3">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="comput_1.html#E7:115_1"><i><font color="Green" title="E84">A6</font></i></a>, <a class="ref" href="comput_1.html#T50" target="_self" title="COMPUT_1:th.50">Th50</a></i>;<br/>
<a NAME="E11:115_1"/><i><font color="Green" title="E85">A7</font></i>: 
(  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="comput_1.html#T35" target="_self" title="COMPUT_1:th.35">Th35</a></i>;<br/>
<a NAME="E12:115_1"/><i><font color="Green" title="E86">A8</font></i>: 
 <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T114" title="FINSEQ_2:th.114">FINSEQ_2:114</a></i>;<br/>
<a NAME="E13:115_1"/><b>then </b><i><font color="Green" title="E87">A9</font></i>: 
( <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E14:115_1"/><i><font color="Green" title="E88">A10</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c4">i</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c4">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_6.html#T61" title="FUNCT_6:th.61">FUNCT_6:61</a></i>;<br/>
<a NAME="E15:115_1"/><i><font color="Green" title="E89">A11</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="txt" href="comput_1.html#E11:115_1"><i><font color="Green" title="E85">A7</font></i></a>, <a class="ref" href="comput_1.html#T26" target="_self" title="COMPUT_1:th.26">Th26</a></i>;<br/>
<a NAME="E16:115_1"/><b>then </b><i><font color="Green" title="E90">A12</font></i>: 
<span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c4">i</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="comput_1.html#E12:115_1"><i><font color="Green" title="E86">A8</font></i></a>, <a class="txt" href="comput_1.html#E13:115_1"><i><font color="Green" title="E87">A9</font></i></a>, <a class="ref" href="funct_6.html#T61" title="FUNCT_6:th.61">FUNCT_6:61</a></i>;<br/>
<a NAME="E17:115_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">1 <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="comput_1.html#D10" target="_self" title="COMPUT_1:def.10">Def10</a></i>;<br/>
<a NAME="E18:115_1"/><b>then </b><i><font color="Green" title="E91">A13</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">g</font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E12:115_1"><i><font color="Green" title="E86">A8</font></i></a>, <a class="txt" href="comput_1.html#E14:115_1"><i><font color="Green" title="E88">A10</font></i></a>, <a class="txt" href="comput_1.html#E15:115_1"><i><font color="Green" title="E89">A11</font></i></a>, <a class="txt" href="comput_1.html#E16:115_1"><i><font color="Green" title="E90">A12</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<a NAME="E19:115_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F</font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E5:115_1"><i><font color="Green" title="E82">A4</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E20:115_1"/><b>then </b>
 <a href="comput_1.html#K2" title="COMPUT_1:func.2">arity</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="comput_1.html#E11:115_1"><i><font color="Green" title="E85">A7</font></i></a>, <a class="ref" href="comput_1.html#D7" target="_self" title="COMPUT_1:def.7">Def7</a></i>;<br/>
<a NAME="E21:115_1"/><b>then </b><i><font color="Green" title="E92">A14</font></i>: 
 <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="comput_1.html#E18:115_1"><i><font color="Green" title="E91">A13</font></i></a>, <a class="ref" href="comput_1.html#T48" target="_self" title="COMPUT_1:th.48">Th48</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>;<br/>
<a NAME="E22:115_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="txt" href="comput_1.html#E11:115_1"><i><font color="Green" title="E85">A7</font></i></a>, <a class="ref" href="comput_1.html#T26" target="_self" title="COMPUT_1:th.26">Th26</a></i>;<br/>
<a NAME="E23:115_1"/><b>then </b>
( <font color="Maroon" title="c9">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c7">F</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> &amp; <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c7">F</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> &amp; <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 1 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="txt" href="comput_1.html#E12:115_1"><i><font color="Green" title="E86">A8</font></i></a>, <a class="txt" href="comput_1.html#E13:115_1"><i><font color="Green" title="E87">A9</font></i></a>, <a class="txt" href="comput_1.html#E14:115_1"><i><font color="Green" title="E88">A10</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a>, <a class="ref" href="funct_6.html#T61" title="FUNCT_6:th.61">FUNCT_6:61</a></i>;<br/>
<b>then </b><font color="Maroon" title="c9">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  = 
<span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="ref" href="comput_1.html#D10" target="_self" title="COMPUT_1:def.10">Def10</a></i>
<br/>.= 
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T25" title="FINSEQ_4:th.25">FINSEQ_4:25</a></i>
;<br/>
<a NAME="E25:115_1"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> 1</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c4">i</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span>
 
<b>by </b><i><a class="txt" href="comput_1.html#E11:115_1"><i><font color="Green" title="E85">A7</font></i></a>, <a class="txt" href="comput_1.html#E13:115_1"><i><font color="Green" title="E87">A9</font></i></a>, <a class="txt" href="comput_1.html#E18:115_1"><i><font color="Green" title="E91">A13</font></i></a>, <a class="txt" href="comput_1.html#E21:115_1"><i><font color="Green" title="E92">A14</font></i></a>, <a class="ref" href="comput_1.html#T52" target="_self" title="COMPUT_1:th.52">Th52</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>;<br/>
<b>hence </b><a NAME="E26:115_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:115"><i><font color="Green" title="E79">A1</font></i></a>, <a class="txt" href="comput_1.html#E5:115_1"><i><font color="Green" title="E82">A4</font></i></a>, <a class="txt" href="comput_1.html#E6:115_1"><i><font color="Green" title="E83">A5</font></i></a>, <a class="txt" href="comput_1.html#E7:115_1"><i><font color="Green" title="E84">A6</font></i></a>, <a class="ref" href="comput_1.html#D15" target="_self" title="COMPUT_1:def.15">Def15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:115"/><i><font color="Green" title="E82">A15</font></i>: 
<font color="Maroon" title="c3">P</font> is <a href="comput_1.html#V6" title="COMPUT_1:attr.6">primitive-recursion_closed</a>
 
<b>by </b><i><a class="ref" href="comput_1.html#D17" target="_self" title="COMPUT_1:def.17">Def17</a></i>;<br/>
<a NAME="E5:115"/><i><font color="Green" title="E83">A16</font></i>: 
for <font color="Olive" title="b1">i</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">i</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="comput_1.html#E2:115"><i><font color="Green" title="E80">A2</font></i></a>, <a class="txt" href="comput_1.html#E3:115"><i><font color="Green" title="E81">A3</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</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>for <font color="Olive" title="b1">c</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  $1 <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>;<br/>
<a NAME="E6:115"/><i><font color="Green" title="E84">A17</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="comput_1.html#E5:115"><i><font color="Green" title="E83">A16</font></i></a></i>;<br/>
<div><i><font color="Green" title="E85">A18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="115_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">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/><b>assume </b><a NAME="E1:115_2"/><i><font color="Green" title="E86">A19</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c4">n</font>]
 ;<br/><b>thus </b><a NAME="E2:115_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">i</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>set </b><font color="Maroon" title="c6">g</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font>;<br/>
<b>set </b><font color="Maroon" title="c7">f1</font> = <font color="Maroon" title="c4">n</font> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font>;<br/>
<b>set </b><font color="Maroon" title="c8">j</font> = <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<b>set </b><font color="Maroon" title="c9">f2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>;<br/>
<a NAME="E1:115_2_1"/><i><font color="Green" title="E87">A20</font></i>: 
 <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="comput_1.html#T35" target="_self" title="COMPUT_1:th.35">Th35</a></i>;<br/>
<a NAME="E2:115_2_1"/><i><font color="Green" title="E88">A21</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E3:115_2_1"/><i><font color="Green" title="E89">A22</font></i>: 
 <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font>
 
<b>by </b><i><a class="ref" href="comput_1.html#T35" target="_self" title="COMPUT_1:th.35">Th35</a></i>;<br/>
<a NAME="E4:115_2_1"/><i><font color="Green" title="E90">A23</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E5:115_2_1"/><i><font color="Green" title="E91">A24</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="comput_1.html#T40" target="_self" title="COMPUT_1:th.40">Th40</a></i>;<br/>
<a NAME="E6:115_2_1"/><i><font color="Green" title="E92">A25</font></i>: 
 <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<b>by </b><i><a class="ref" href="comput_1.html#T41" target="_self" title="COMPUT_1:th.41">Th41</a></i>;<br/>
<a NAME="E7:115_2_1"/><i><font color="Green" title="E93">A26</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</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">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E8:115_2_1"/><i><font color="Green" title="E94">A27</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font> <a href="comput_1.html#R1" title="COMPUT_1:pred.1">is_primitive-recursively_expressed_by</a> <font color="Maroon" title="c4">n</font> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_2_1_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon" title="c10">m</font> =  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="comput_1.html#D12" target="_self" title="COMPUT_1:def.12">COMPUT_1:def 12</a><br/>

<b>thus </b><a NAME="E1:115_2_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">m</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="comput_1.html#T24" target="_self" title="COMPUT_1:th.24">Th24</a></i>;<br/>

<b>thus </b><a NAME="E2:115_2_1_1"/>
( <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font> )
 <b>by </b><i><a class="ref" href="comput_1.html#T35" target="_self" title="COMPUT_1:th.35">Th35</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>

<b>thus </b><a NAME="E3:115_2_1_1"/>
( <span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m</font> &amp; <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="comput_1.html#E3:115_2_1"><i><font color="Green" title="E89">A22</font></i></a>, <a class="txt" href="comput_1.html#E6:115_2_1"><i><font color="Green" title="E92">A25</font></i></a>, <a class="txt" href="comput_1.html#E7:115_2_1"><i><font color="Green" title="E93">A26</font></i></a>, <a class="ref" href="comput_1.html#T35" target="_self" title="COMPUT_1:th.35">Th35</a></i>;<br/>

<b>let </b><font color="Maroon" title="c11">p</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E4:115_2_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m</font>
 ;<br/>

<a NAME="E5:115_2_1_1"/><b>then </b><i><font color="Green" title="E95">A28</font></i>: 
( <font color="Maroon" title="c11">p</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="comput_1.html#E1:115_2_1"><i><font color="Green" title="E87">A20</font></i></a>, <a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E6:115_2_1_1"/><b>then </b><i><font color="Green" title="E96">A29</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c11">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font>
 
<b>by </b><i><a class="ref" href="comput_1.html#T12" target="_self" title="COMPUT_1:th.12">Th12</a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>thus </b><a NAME="E7:115_2_1_1"/>
( <font color="Maroon" title="c11">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font></span>)</span> implies  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c11">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="comput_1.html#E4:115_2_1"><i><font color="Green" title="E90">A23</font></i></a>, <a class="txt" href="comput_1.html#E5:115_2_1_1"><i><font color="Green" title="E95">A28</font></i></a>, <a class="ref" href="comput_1.html#T12" target="_self" title="COMPUT_1:th.12">Th12</a></i>;<br/>

<b>thus </b><a NAME="E8:115_2_1_1"/>
(  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c11">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font></span>)</span> implies <font color="Maroon" title="c11">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="comput_1.html#E2:115_2_1"><i><font color="Green" title="E88">A21</font></i></a>, <a class="txt" href="comput_1.html#E5:115_2_1_1"><i><font color="Green" title="E95">A28</font></i></a>, <a class="txt" href="comput_1.html#E94"><i><font color="Green" title="E75">Lm7</font></i></a></i>;<br/>

<b>thus </b><a NAME="E9:115_2_1_1"/>
( <font color="Maroon" title="c11">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c11">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="comput_1.html#E5:115_2_1_1"><i><font color="Green" title="E95">A28</font></i></a>, <a class="txt" href="comput_1.html#E6:115_2_1_1"><i><font color="Green" title="E96">A29</font></i></a>, <a class="txt" href="comput_1.html#E94"><i><font color="Green" title="E75">Lm7</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>

<b>let </b><font color="Maroon" title="c12">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E10:115_2_1_1"/><i><font color="Green" title="E97">A30</font></i>: 
<font color="Maroon" title="c11">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="txt" href="comput_1.html#E5:115_2_1_1"><i><font color="Green" title="E95">A28</font></i></a>, <a class="txt" href="comput_1.html#E94"><i><font color="Green" title="E75">Lm7</font></i></a></i>;<br/>
<a NAME="E11:115_2_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E12:115_2_1_1"/><b>then </b><i><font color="Green" title="E98">A31</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c5">i</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>;<br/>
<a NAME="E13:115_2_1_1"/><i><font color="Green" title="E99">A32</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c5">i</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="txt" href="comput_1.html#E7:115_2_1"><i><font color="Green" title="E93">A26</font></i></a>, <a class="txt" href="comput_1.html#E10:115_2_1_1"><i><font color="Green" title="E97">A30</font></i></a>, <a class="ref" href="finseq_2.html#T127" title="FINSEQ_2:th.127">FINSEQ_2:127</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:115_2_1_1_1_1"/>
<font color="Maroon" title="c11">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font></span>)</span>
 ;<br/><b>thus </b><a NAME="E2:115_2_1_1_1_1"/>
<font color="Maroon" title="c11">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E2:115_2_1"><i><font color="Green" title="E88">A21</font></i></a>, <a class="txt" href="comput_1.html#E5:115_2_1_1"><i><font color="Green" title="E95">A28</font></i></a>, <a class="txt" href="comput_1.html#E94"><i><font color="Green" title="E75">Lm7</font></i></a></i>;<br/><a NAME="E3:115_2_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E5:115_2_1_1"><i><font color="Green" title="E95">A28</font></i></a>, <a class="txt" href="comput_1.html#E94"><i><font color="Green" title="E75">Lm7</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E4:115_2_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">m</font></span>)</span></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E5:115_2_1"><i><font color="Green" title="E91">A24</font></i></a>, <a class="txt" href="comput_1.html#E13:115_2_1_1"><i><font color="Green" title="E99">A32</font></i></a></i>;<br/></div>
<b>end;</b></div><b>thus </b><a NAME="E2:115_2_1_1_1"/>
( <font color="Maroon" title="c11">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">m</font></span>)</span></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> implies <font color="Maroon" title="c11">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="comput_1.html#E2:115_2_1"><i><font color="Green" title="E88">A21</font></i></a>, <a class="txt" href="comput_1.html#E5:115_2_1_1"><i><font color="Green" title="E95">A28</font></i></a>, <a class="txt" href="comput_1.html#E94"><i><font color="Green" title="E75">Lm7</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E15:115_2_1_1"/>
<font color="Maroon" title="c11">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font></span>)</span>
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> = 
<font color="Maroon" title="c5">i</font>
<b>by </b><i><a class="txt" href="comput_1.html#E5:115_2_1_1"><i><font color="Green" title="E95">A28</font></i></a>, <a class="txt" href="comput_1.html#E94"><i><font color="Green" title="E75">Lm7</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c5">i</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="comput_1.html#E12:115_2_1_1"><i><font color="Green" title="E98">A31</font></i></a>, <a class="txt" href="comput_1.html#E13:115_2_1_1"><i><font color="Green" title="E99">A32</font></i></a>, <a class="ref" href="comput_1.html#T42" target="_self" title="COMPUT_1:th.42">Th42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">m</font></span>)</span></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="comput_1.html#E5:115_2_1_1"><i><font color="Green" title="E95">A28</font></i></a>, <a class="txt" href="comput_1.html#E94"><i><font color="Green" title="E75">Lm7</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E9:115_2_1"/><i><font color="Green" title="E95">A33</font></i>: 
( <font color="Maroon" title="c4">n</font> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font> is    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K3" title="COMPUT_1:func.3">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> is    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K3" title="COMPUT_1:func.3">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font> is    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K3" title="COMPUT_1:func.3">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="comput_1.html#T31" target="_self" title="COMPUT_1:th.31">Th31</a></i>;<br/>
<a NAME="E10:115_2_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<b>by </b><i><a class="txt" href="comput_1.html#E7:115_2_1"><i><font color="Green" title="E93">A26</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E11:115_2_1"/><b>then </b>
( <font color="Maroon" title="c4">n</font> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> )
 
<b>by </b><i><a class="txt" href="comput_1.html#E1:115_2"><i><font color="Green" title="E86">A19</font></i></a>, <a class="ref" href="comput_1.html#D17" target="_self" title="COMPUT_1:def.17">Def17</a></i>;<br/>
<b>hence </b><a NAME="E12:115_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E4:115"><i><font color="Green" title="E82">A15</font></i></a>, <a class="txt" href="comput_1.html#E8:115_2_1"><i><font color="Green" title="E94">A27</font></i></a>, <a class="txt" href="comput_1.html#E9:115_2_1"><i><font color="Green" title="E95">A33</font></i></a>, <a class="ref" href="comput_1.html#D16" target="_self" title="COMPUT_1:def.16">Def16</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E8:115"/>
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">S<sub>3</sub></font>[<font color="Olive" title="b1">n</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="comput_1.html#E6:115"><i><font color="Green" title="E84">A17</font></i></a>, <a class="txt" href="comput_1.html#E7:115"><i><font color="Green" title="E85">A18</font></i></a>);<br/></i>
<b>hence </b><a NAME="E9:115"/>
<font color="Maroon" title="c1">n</font> <a href="comput_1.html#K4" title="COMPUT_1:func.4">const</a> <font color="Maroon" title="c2">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 ;<br/>


</div>
