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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="comput_1.html#V8" target="_self">primitive-recursively_closed</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> </span>)</span>;<br/>





<a NAME="E1:118"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="comput_1.html#V6" target="_self">composition_closed</a>
 
<b>by </b><i><a class="ref" href="comput_1.html#D17" target="_self">Def17</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means 0 <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E2:118"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="ref" href="comput_1.html#D17" target="_self">Def17</a></i>;<br/>
<a NAME="E3:118"/><i><font color="Green">E84</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>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</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="118_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default">0 <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = 0 <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>4</sub></font> as    <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="comput_1.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E2:118_1"/>
<span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default">0 <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> as  <a href="comput_1.html#V5" target="_self">with_the_same_arity</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>assume </b><a NAME="E4:118_1"/>
0 <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E5:118_1"/><b>then </b><i><font color="Green">E85</font></i>: 
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default">0 <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default">0 <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E6:118_1"/><i><font color="Green">E86</font></i>: 
1 <a href="comput_1.html#K7" target="_self">succ</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="comput_1.html#D17" target="_self">Def17</a></i>;<br/>
<i><font color="Green">E87</font></i>:  <a href="comput_1.html#K3" target="_self">arity</a> <span class="p1">(<span class="default">1 <a href="comput_1.html#K7" target="_self">succ</a> 1</span>)</span> = 
1
<b>by </b><i><a class="ref" href="comput_1.html#T38" target="_self">Th38</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = 1 <a href="comput_1.html#K7" target="_self">succ</a> 1 as  <a href="unialg_1.html#V2">quasi_total</a>  <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="comput_1.html#T32" target="_self">Th32</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:118_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E2:118_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="comput_1.html#E5:118_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:118_1_2"/>
<font color="Maroon">c<sub>9</sub></font> is <a href="unialg_1.html#V2">quasi_total</a>
 <b>by </b><i><a class="ref" href="comput_1.html#T32" target="_self">Th32</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="funct_6.html#K6">:&gt;</a></span> as  <a href="unialg_1.html#V2">quasi_total</a>  <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="comput_1.html#E7:118_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="comput_1.html#T50" target="_self">Th50</a></i>;<br/>
<a NAME="E11:118_1"/><i><font color="Green">E88</font></i>: 
(  <a href="comput_1.html#K3" target="_self">arity</a> <span class="p1">(<span class="default">0 <a href="comput_1.html#K6" target="_self">const</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="comput_1.html#K3" target="_self">arity</a> <span class="p1">(<span class="default">0 <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="comput_1.html#T35" target="_self">Th35</a></i>;<br/>
<a NAME="E12:118_1"/><i><font color="Green">E89</font></i>: 
 <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K5">NAT</a>  is    <a href="finseq_2.html#M2">Element</a> of 0 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T114">FINSEQ_2:114</a></i>;<br/>
<a NAME="E13:118_1"/><b>then </b><i><font color="Green">E90</font></i>: 
( <span class="p1">(<span class="default">0 <a href="comput_1.html#K6" target="_self">const</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default">0 <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="comput_1.html#T36" target="_self">Th36</a></i>;<br/>
<a NAME="E14:118_1"/><i><font color="Green">E91</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_6.html#K6">&lt;:</a><span class="default"><span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default">0 <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="funct_6.html#K6">:&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0 <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_6.html#T61">FUNCT_6:61</a></i>;<br/>
<a NAME="E15:118_1"/><i><font color="Green">E92</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0 <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="txt" href="comput_1.html#E11:118_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="comput_1.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E16:118_1"/><b>then </b><i><font color="Green">E93</font></i>: 
<span class="p1"><a href="funct_6.html#K6">&lt;:</a><span class="default"><span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default">0 <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="funct_6.html#K6">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="comput_1.html#E12:118_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="comput_1.html#E13:118_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="funct_6.html#T61">FUNCT_6:61</a></i>;<br/>
<a NAME="E17:118_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">1 <a href="comput_1.html#K7" target="_self">succ</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="comput_1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E18:118_1"/><b>then </b><i><font color="Green">E94</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E12:118_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="comput_1.html#E14:118_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="comput_1.html#E15:118_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="comput_1.html#E16:118_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>
<a NAME="E19:118_1"/>
0 <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E5:118_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E20:118_1"/><b>then </b>
 <a href="comput_1.html#K4" target="_self">arity</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="comput_1.html#E11:118_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="comput_1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E21:118_1"/><b>then </b><i><font color="Green">E95</font></i>: 
 <a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="comput_1.html#E18:118_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="comput_1.html#T48" target="_self">Th48</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>
<a NAME="E22:118_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0 <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="txt" href="comput_1.html#E11:118_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="comput_1.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E23:118_1"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="comput_1.html#K7" target="_self">succ</a> 1</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="funct_6.html#K6">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> &amp; <span class="p1"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="funct_6.html#K6">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> &amp; <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> is    <a href="finseq_2.html#M2">Element</a> of 1 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="txt" href="comput_1.html#E12:118_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="comput_1.html#E13:118_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="comput_1.html#E14:118_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a>, <a class="ref" href="funct_6.html#T61">FUNCT_6:61</a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  = 
<span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="comput_1.html#D10" target="_self">Def10</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>
;<br/>
<a NAME="E25:118_1"/><b>then </b>
0 <a href="comput_1.html#K6" target="_self">const</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="comput_1.html#K7" target="_self">succ</a> 1</span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1"><a href="funct_6.html#K6">&lt;:</a><span class="default"><span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default">0 <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="funct_6.html#K6">:&gt;</a></span>
 
<b>by </b><i><a class="txt" href="comput_1.html#E11:118_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="comput_1.html#E13:118_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="comput_1.html#E18:118_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="comput_1.html#E21:118_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="comput_1.html#T52" target="_self">Th52</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>
<b>hence </b><a NAME="E26:118_1"/>
0 <a href="comput_1.html#K6" target="_self">const</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:118"><i><font color="Green">E82</font></i></a>, <a class="txt" href="comput_1.html#E5:118_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="comput_1.html#E6:118_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="comput_1.html#E7:118_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="comput_1.html#D15" target="_self">Def15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:118"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="comput_1.html#V7" target="_self">primitive-recursion_closed</a>
 
<b>by </b><i><a class="ref" href="comput_1.html#D17" target="_self">Def17</a></i>;<br/>
<a NAME="E5:118"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>2</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="comput_1.html#E2:118"><i><font color="Green">E83</font></i></a>, <a class="txt" href="comput_1.html#E3:118"><i><font color="Green">E84</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">a<sub>1</sub></font> <a href="comput_1.html#K6" target="_self">const</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E6:118"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<b>by </b><i><a class="txt" href="comput_1.html#E5:118"><i><font color="Green">E86</font></i></a></i>;<br/>
<div><i><font color="Green">E88</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="118_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:118_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 ;<br/><b>thus </b><a NAME="E2:118_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="comput_1.html#K8" target="_self">proj</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span>;<br/>
<a NAME="E1:118_2_1"/><i><font color="Green">E90</font></i>: 
 <a href="comput_1.html#K3" target="_self">arity</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="comput_1.html#T35" target="_self">Th35</a></i>;<br/>
<a NAME="E2:118_2_1"/><i><font color="Green">E91</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E3:118_2_1"/><i><font color="Green">E92</font></i>: 
 <a href="comput_1.html#K3" target="_self">arity</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="comput_1.html#T35" target="_self">Th35</a></i>;<br/>
<a NAME="E4:118_2_1"/><i><font color="Green">E93</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E5:118_2_1"/><i><font color="Green">E94</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="comput_1.html#K8" target="_self">proj</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="comput_1.html#T40" target="_self">Th40</a></i>;<br/>
<a NAME="E6:118_2_1"/><i><font color="Green">E95</font></i>: 
 <a href="comput_1.html#K3" target="_self">arity</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="comput_1.html#K8" target="_self">proj</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="comput_1.html#T41" target="_self">Th41</a></i>;<br/>
<a NAME="E7:118_2_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <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>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<a NAME="E8:118_2_1"/><i><font color="Green">E97</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font> <a href="comput_1.html#R1" target="_self">is_primitive-recursively_expressed_by</a> <font color="Maroon">c<sub>4</sub></font> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="comput_1.html#K8" target="_self">proj</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_2_1_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">c<sub>10</sub></font> =  <a href="comput_1.html#K3" target="_self">arity</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="comput_1.html#D12" target="_self">COMPUT_1:def 12</a><br/>

<b>thus </b><a NAME="E1:118_2_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="comput_1.html#T24" target="_self">Th24</a></i>;<br/>

<b>thus </b><a NAME="E2:118_2_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="comput_1.html#T35" target="_self">Th35</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>

<b>thus </b><a NAME="E3:118_2_1_1"/>
( <span class="p1">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K3" target="_self">arity</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="comput_1.html#K8" target="_self">proj</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="comput_1.html#E3:118_2_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="comput_1.html#E6:118_2_1"><i><font color="Green">E95</font></i></a>, <a class="txt" href="comput_1.html#E7:118_2_1"><i><font color="Green">E96</font></i></a>, <a class="ref" href="comput_1.html#T35" target="_self">Th35</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E4:118_2_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E5:118_2_1_1"/><b>then </b><i><font color="Green">E98</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="comput_1.html#E1:118_2_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E6:118_2_1_1"/><b>then </b>
 <a href="finseq_3.html#K2">Del</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="comput_1.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E7:118_2_1_1"/><b>then </b><i><font color="Green">E99</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="comput_1.html#T36" target="_self">Th36</a></i>;<br/>
<b>thus </b><a NAME="E8:118_2_1_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="funct_7.html#K3">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> implies  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="comput_1.html#E4:118_2_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="comput_1.html#E5:118_2_1_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="comput_1.html#T12" target="_self">Th12</a></i>;<br/>

<b>thus </b><a NAME="E9:118_2_1_1"/>
(  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> implies <font color="Maroon">c<sub>11</sub></font> <a href="funct_7.html#K3">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="comput_1.html#E2:118_2_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="comput_1.html#E5:118_2_1_1"><i><font color="Green">E98</font></i></a>, <a class="txt" href="comput_1.html#E94"><i><font color="Green">Lemma78</font></i></a></i>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:118_2_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_7.html#K3">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/><a NAME="E2:118_2_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_7.html#K3">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,0 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E5:118_2_1_1"><i><font color="Green">E98</font></i></a>, <a class="txt" href="comput_1.html#E94"><i><font color="Green">Lemma78</font></i></a></i>;<br/><b>hence </b><a NAME="E3:118_2_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_7.html#K3">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E7:118_2_1_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="comput_1.html#T36" target="_self">Th36</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E11:118_2_1_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_7.html#K3">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="txt" href="comput_1.html#E5:118_2_1_1"><i><font color="Green">E98</font></i></a>, <a class="txt" href="comput_1.html#E94"><i><font color="Green">Lemma78</font></i></a></i>;<br/>
<a NAME="E12:118_2_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_7.html#K3">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E13:118_2_1_1"/><b>then </b><i><font color="Green">E101</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_7.html#K3">+*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</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">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/>
<a NAME="E14:118_2_1_1"/><i><font color="Green">E102</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_7.html#K3">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="txt" href="comput_1.html#E7:118_2_1"><i><font color="Green">E96</font></i></a>, <a class="txt" href="comput_1.html#E11:118_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="finseq_2.html#T127">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="E1:118_2_1_1_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_7.html#K3">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/><b>thus </b><a NAME="E2:118_2_1_1_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_7.html#K3">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E2:118_2_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="comput_1.html#E5:118_2_1_1"><i><font color="Green">E98</font></i></a>, <a class="txt" href="comput_1.html#E94"><i><font color="Green">Lemma78</font></i></a></i>;<br/><a NAME="E3:118_2_1_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_7.html#K3">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="comput_1.html#E11:118_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="comput_1.html#T36" target="_self">Th36</a></i>;<br/><b>hence </b><a NAME="E4:118_2_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_7.html#K3">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&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">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_7.html#K3">+*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="comput_1.html#K8" target="_self">proj</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E5:118_2_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="comput_1.html#E14:118_2_1_1"><i><font color="Green">E102</font></i></a></i>;<br/></div>
<b>end;</b></div><b>thus </b><a NAME="E2:118_2_1_1_2"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="funct_7.html#K3">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_7.html#K3">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&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">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_7.html#K3">+*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="comput_1.html#K8" target="_self">proj</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> implies <font color="Maroon">c<sub>11</sub></font> <a href="funct_7.html#K3">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="comput_1.html#E2:118_2_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="comput_1.html#E5:118_2_1_1"><i><font color="Green">E98</font></i></a>, <a class="txt" href="comput_1.html#E94"><i><font color="Green">Lemma78</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E16:118_2_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_7.html#K3">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/>

<a NAME="E17:118_2_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_7.html#K3">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="txt" href="comput_1.html#E5:118_2_1_1"><i><font color="Green">E98</font></i></a>, <a class="txt" href="comput_1.html#E94"><i><font color="Green">Lemma78</font></i></a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_7.html#K3">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="comput_1.html#T36" target="_self">Th36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="comput_1.html#K8" target="_self">proj</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_7.html#K3">+*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="comput_1.html#E13:118_2_1_1"><i><font color="Green">E101</font></i></a>, <a class="txt" href="comput_1.html#E14:118_2_1_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="comput_1.html#T42" target="_self">Th42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="comput_1.html#K8" target="_self">proj</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_7.html#K3">+*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&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">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_7.html#K3">+*</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="comput_1.html#E11:118_2_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="comput_1.html#T36" target="_self">Th36</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E9:118_2_1"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font> is    <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="comput_1.html#K8" target="_self">proj</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> is    <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font> is    <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="comput_1.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E10:118_2_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="txt" href="comput_1.html#E7:118_2_1"><i><font color="Green">E96</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E11:118_2_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="comput_1.html#K8" target="_self">proj</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="comput_1.html#E1:118_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="comput_1.html#D17" target="_self">Def17</a></i>;<br/>
<b>hence </b><a NAME="E12:118_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="comput_1.html#E4:118"><i><font color="Green">E85</font></i></a>, <a class="txt" href="comput_1.html#E8:118_2_1"><i><font color="Green">E97</font></i></a>, <a class="txt" href="comput_1.html#E9:118_2_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="comput_1.html#D16" target="_self">Def16</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E8:118"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>3</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="comput_1.html#E6:118"><i><font color="Green">E87</font></i></a>, <a class="txt" href="comput_1.html#E7:118"><i><font color="Green">E88</font></i></a>);<br/></i>
<b>hence </b><a NAME="E9:118"/>
<font color="Maroon">c<sub>1</sub></font> <a href="comput_1.html#K6" target="_self">const</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div>
