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

<b>set </b><font color="Maroon">S</font> = <span class="p1">{<span class="default"> <font color="Olive">R</font> where <font color="Olive">R</font> is   <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> : <font color="Olive">R</font> is <a href="comput_1.html#V8" target="_self">primitive-recursively_closed</a> </span>}</span> ;<br/>
<a NAME="E1:125_1_1"/>
(  <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a>  <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"><a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> </span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> </span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <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> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="zfmisc_1.html#T80">ZFMISC_1:80</a></i>;<br/>
<a NAME="E2:125_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">R</font> where <font color="Olive">R</font> is   <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> : <font color="Olive">R</font> is <a href="comput_1.html#V8" target="_self">primitive-recursively_closed</a> </span>}</span> 
 
<b>by </b><i><a class="ref" href="comput_1.html#D1" target="_self">Def1</a></i>;<br/>
<div><i><font color="Green">E49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:125_1_1_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">R</font> where <font color="Olive">R</font> is   <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> : <font color="Olive">R</font> is <a href="comput_1.html#V8" target="_self">primitive-recursively_closed</a> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">R</font> being   <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><b> such that </b><br/><a NAME="E3:125_1_1_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">R</font> is <a href="comput_1.html#V8" target="_self">primitive-recursively_closed</a> )
 ;<br/><b>thus </b><a NAME="E4:125_1_1_1"/>
0 <a href="comput_1.html#K6" target="_self">const</a> 0 <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="comput_1.html#T18" target="_self">Th18</a>, <a class="ref" href="comput_1.html#T13" target="_self">Th13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:125_1_1"/>
not  <a href="comput_1.html#K11" target="_self">PrimRec</a>  is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="comput_1.html#T17" target="_self">Th17</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>

<b>thus </b><a NAME="E5:125_1_1"/>
 <a href="comput_1.html#K11" target="_self">PrimRec</a>  is <a href="comput_1.html#V8" target="_self">primitive-recursively_closed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:125_1_1_2"/>
0 <a href="comput_1.html#K6" target="_self">const</a> 0 <a href="hidden.html#R2">in</a>  <a href="comput_1.html#K11" target="_self">PrimRec</a> 
 <b>by </b><i><a class="ref" href="comput_1.html#T17" target="_self">Th17</a>, , <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="comput_1.html#D17" target="_self">COMPUT_1:def 17</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:125_1_1_2_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">R</font> where <font color="Olive">R</font> is   <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> : <font color="Olive">R</font> is <a href="comput_1.html#V8" target="_self">primitive-recursively_closed</a> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">R</font> being   <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><b> such that </b><br/><a NAME="E3:125_1_1_2_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">R</font> is <a href="comput_1.html#V8" target="_self">primitive-recursively_closed</a> )
 ;<br/><b>thus </b><a NAME="E4:125_1_1_2_1"/>
1 <a href="comput_1.html#K7" target="_self">succ</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="comput_1.html#T18" target="_self">Th18</a>, <a class="ref" href="comput_1.html#T13" target="_self">Th13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:125_1_1_2"/>
1 <a href="comput_1.html#K7" target="_self">succ</a> 1 <a href="hidden.html#R2">in</a>  <a href="comput_1.html#K11" target="_self">PrimRec</a> 
 <b>by </b><i><a class="ref" href="comput_1.html#T17" target="_self">Th17</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>

<div><b>hereby </b>
<div class="add"><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> , <font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:125_1_1_2_2"/><i><font color="Green">E53</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="125_1_1_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:125_1_1_2_2_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">R</font> where <font color="Olive">R</font> is   <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> : <font color="Olive">R</font> is <a href="comput_1.html#V8" target="_self">primitive-recursively_closed</a> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">R</font> being   <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><b> such that </b><br/><a NAME="E3:125_1_1_2_2_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">R</font> is <a href="comput_1.html#V8" target="_self">primitive-recursively_closed</a> )
 ;<br/><b>thus </b><a NAME="E4:125_1_1_2_2_1"/>
<font color="Maroon">n</font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="comput_1.html#T18" target="_self">Th18</a>, , <a class="ref" href="comput_1.html#T13" target="_self">Th13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:125_1_1_2_2"/>
<font color="Maroon">n</font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="comput_1.html#K11" target="_self">PrimRec</a> 
 <b>by </b><i><a class="ref" href="comput_1.html#T17" target="_self">Th17</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="comput_1.html#D15" target="_self">COMPUT_1:def 15</a>
<div class="add"><b>let </b><font color="Maroon">f</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>let </b><font color="Maroon">F</font> be  <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="E1:125_1_1_2_3"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="comput_1.html#K11" target="_self">PrimRec</a>  &amp;  <a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="comput_1.html#K11" target="_self">PrimRec</a>  )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_1_2_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:125_1_1_2_3_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">R</font> where <font color="Olive">R</font> is   <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> : <font color="Olive">R</font> is <a href="comput_1.html#V8" target="_self">primitive-recursively_closed</a> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">R</font> being   <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><b> such that </b><br/><a NAME="E3:125_1_1_2_3_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">R</font> is <a href="comput_1.html#V8" target="_self">primitive-recursively_closed</a> )
 ;<br/><a NAME="E4:125_1_1_2_3_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">R</font> is <a href="comput_1.html#V6" target="_self">composition_closed</a>
 <b>by </b><i><a class="ref" href="comput_1.html#D2" target="_self">Def2</a>, <a class="ref" href="comput_1.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E5:125_1_1_2_3_1"/>
 <a href="comput_1.html#K11" target="_self">PrimRec</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="comput_1.html#D2" target="_self">Def2</a>, <a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/><a NAME="E6:125_1_1_2_3_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> )
 <b>by </b><i><a class="ref" href="comput_1.html#T18" target="_self">Th18</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E7:125_1_1_2_3_1"/>
<font color="Maroon">f</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">F</font></span><a href="funct_6.html#K6">:&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="comput_1.html#T18" target="_self">Th18</a>, <a class="ref" href="comput_1.html#D2" target="_self">Def2</a>, <a class="ref" href="comput_1.html#D3" target="_self">Def3</a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:125_1_1_2_3"/>
<font color="Maroon">f</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">F</font></span><a href="funct_6.html#K6">:&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="comput_1.html#K11" target="_self">PrimRec</a> 
 <b>by </b><i><a class="ref" href="comput_1.html#T17" target="_self">Th17</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="comput_1.html#D16" target="_self">COMPUT_1:def 16</a>
<div class="add"><b>let </b><font color="Maroon">g</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> , <font color="Maroon">f1</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> , <font color="Maroon">f2</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>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>assume </b><b>that </b><br/><a NAME="E1:125_1_1_2_4"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">g</font> <a href="comput_1.html#R1" target="_self">is_primitive-recursively_expressed_by</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E2:125_1_1_2_4"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">f1</font> <a href="hidden.html#R2">in</a>  <a href="comput_1.html#K11" target="_self">PrimRec</a> 
 <b>and </b><br/><a NAME="E3:125_1_1_2_4"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">f2</font> <a href="hidden.html#R2">in</a>  <a href="comput_1.html#K11" target="_self">PrimRec</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_1_2_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:125_1_1_2_4_1"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">R</font> where <font color="Olive">R</font> is   <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> : <font color="Olive">R</font> is <a href="comput_1.html#V8" target="_self">primitive-recursively_closed</a> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">R</font> being   <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><b> such that </b><br/><a NAME="E3:125_1_1_2_4_1"/><i><font color="Green">E137</font></i>: 
( <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">R</font> is <a href="comput_1.html#V8" target="_self">primitive-recursively_closed</a> )
 ;<br/><a NAME="E4:125_1_1_2_4_1"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">R</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#D4" target="_self">Def4</a>, <a class="ref" href="comput_1.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E5:125_1_1_2_4_1"/>
( <font color="Maroon">f1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">f2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 <b>by </b><i>, <a class="ref" href="comput_1.html#D2" target="_self">Def2</a>, <a class="ref" href="comput_1.html#D3" target="_self">Def3</a>, <a class="ref" href="comput_1.html#D4" target="_self">Def4</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><b>hence </b><a NAME="E6:125_1_1_2_4_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="comput_1.html#T18" target="_self">Th18</a>, <a class="ref" href="comput_1.html#D4" target="_self">Def4</a>, <a class="ref" href="comput_1.html#T19" target="_self">Th19</a>, <a class="ref" href="comput_1.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:125_1_1_2_4"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="comput_1.html#K11" target="_self">PrimRec</a> 
 <b>by </b><i><a class="ref" href="comput_1.html#T17" target="_self">Th17</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/></div>
<b>end;</b></div>


</div><b>end;</b></div>


</div>
