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

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">P</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/>





<b>assume </b><a NAME="E1:119"/><i><font color="Green">E48</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:119_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="comput_1.html#K7" target="_self">succ</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i/>;<br/><a NAME="E2:119_1"/><i><font color="Green">E49</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i/>;<br/><a NAME="E3:119_1"/><b>then </b><i><font color="Green">E51</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"><font color="Maroon">n</font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Maroon">i</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> <font color="Maroon">n</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="funct_6.html#T61">FUNCT_6:61</a></i>;<br/><a NAME="E4:119_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="comput_1.html#T15" target="_self">Th15</a>, </i>;<br/><a NAME="E5:119_1"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="relat_1.html#K2">rng</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"><font color="Maroon">n</font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Maroon">i</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> 1 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i/>;<br/><a NAME="E6:119_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/>;<br/><b>hence </b><a NAME="E7:119_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2"><a href="funct_6.html#K6">&lt;:</a><span class="default"><span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Maroon">i</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="funct_6.html#K6">:&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</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#T16" target="_self">Th16</a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E8:119_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E10:119_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/><a NAME="E11:119_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x'</font>
 <b>by </b><i><a class="ref" href="comput_1.html#T15" target="_self">Th15</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>set </b><font color="Maroon">xi</font> = <font color="Maroon">x'</font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">i</font>;<br/><i><font color="Green">E54</font></i>: <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="comput_1.html#K7" target="_self">succ</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="comput_1.html#T17" target="_self">Th17</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><span class="p1">(<span class="default"><span class="p2">(<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="p2"><a href="funct_6.html#K6">&lt;:</a><span class="default"><span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Maroon">i</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="funct_6.html#K6">:&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> = 
<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"><span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Maroon">i</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> <font color="Maroon">x'</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<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"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i>, <a class="ref" href="funct_6.html#T61">FUNCT_6:61</a></i>
<br/>.= 
<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"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x'</font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span><a href="binarith.html#K11">*&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/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <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/><b>hence </b><a NAME="E14:119_1"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="comput_1.html#K7" target="_self">succ</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2"><a href="funct_6.html#K6">&lt;:</a><span class="default"><span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Maroon">i</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span><a href="funct_6.html#K6">:&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E3:119"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">n</font> <a href="comput_1.html#K7" target="_self">succ</a> <font color="Maroon">i</font> <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"><font color="Maroon">n</font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Maroon">i</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="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E4:119"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">P</font> is <a href="comput_1.html#V6" target="_self">composition_closed</a>
 
<b>by </b><i><a class="ref" href="comput_1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E5:119"/><i><font color="Green">E58</font></i>: 
1 <a href="comput_1.html#K7" target="_self">succ</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="comput_1.html#T13" target="_self">Th13</a></i>;<br/>
<i><font color="Green">E64</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/>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Maroon">i</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/>
<a NAME="E7:119"/><i><font color="Green">E65</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Maroon">i</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:119_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Maroon">i</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
 ;<br/>

<a NAME="E2:119_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Maroon">i</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></i>;<br/>
<a NAME="E3:119_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:119_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="comput_1.html#T15" target="_self">Th15</a>, <a class="ref" href="comput_1.html#T13" target="_self">Th13</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">nproji</font> = <font color="Maroon">n</font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Maroon">i</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/>;<br/>
<a NAME="E9:119"/>
<span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">nproji</font></span><a href="binarith.html#K11">*&gt;</a></span> is  <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>hence </b><a NAME="E10:119"/>
<font color="Maroon">n</font> <a href="comput_1.html#K7" target="_self">succ</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="comput_1.html#T16" target="_self">Th16</a>, <a class="ref" href="comput_1.html#T17" target="_self">Th17</a>, , </i>;<br/>


</div>
