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

<b>set </b><font color="Maroon" title="c1">S</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">R</font> where <font color="Olive" title="b1">R</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> : <font color="Olive" title="b1">R</font> is <a href="comput_1.html#V7" title="COMPUT_1:attr.7">primitive-recursively_closed</a> </span>}</span> ;<br/>
<a NAME="E1:123_1_1"/>
(  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="zfmisc_1.html#T80" title="ZFMISC_1:th.80">ZFMISC_1:80</a></i>;<br/>
<a NAME="E2:123_1_1"/><b>then </b><i><font color="Green" title="E84">A1</font></i>: 
 <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">R</font> where <font color="Olive" title="b1">R</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> : <font color="Olive" title="b1">R</font> is <a href="comput_1.html#V7" title="COMPUT_1:attr.7">primitive-recursively_closed</a> </span>}</span> 
 
<b>by </b><i><a class="ref" href="comput_1.html#T70" target="_self" title="COMPUT_1:th.70">Th70</a></i>;<br/>
<div><i><font color="Green" title="E85">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="123_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:123_1_1_1"/>
<font color="Maroon" title="c2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">R</font> where <font color="Olive" title="b1">R</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> : <font color="Olive" title="b1">R</font> is <a href="comput_1.html#V7" title="COMPUT_1:attr.7">primitive-recursively_closed</a> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c3">R</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span><b> such that </b><br/><a NAME="E3:123_1_1_1"/><i><font color="Green" title="E86">A3</font></i>: 
( <font color="Maroon" title="c3">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c3">R</font> is <a href="comput_1.html#V7" title="COMPUT_1:attr.7">primitive-recursively_closed</a> )
 ;<br/><b>thus </b><a NAME="E4:123_1_1_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K5" title="COMPUT_1:func.5">const</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:123_1_1_1"><i><font color="Green" title="E86">A3</font></i></a>, <a class="ref" href="comput_1.html#D17" target="_self" title="COMPUT_1:def.17">Def17</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:123_1_1"/>
not  <a href="comput_1.html#K10" title="COMPUT_1:func.10">PrimRec</a>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="comput_1.html#E2:123_1_1"><i><font color="Green" title="E84">A1</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>

<b>thus </b><a NAME="E5:123_1_1"/>
 <a href="comput_1.html#K10" title="COMPUT_1:func.10">PrimRec</a>  is <a href="comput_1.html#V7" title="COMPUT_1:attr.7">primitive-recursively_closed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="123_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:123_1_1_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K5" title="COMPUT_1:func.5">const</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K10" title="COMPUT_1:func.10">PrimRec</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E2:123_1_1"><i><font color="Green" title="E84">A1</font></i></a>, <a class="txt" href="comput_1.html#E3:123_1_1"><i><font color="Green" title="E85">A2</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="comput_1.html#D17" target="_self" title="COMPUT_1:def.17">COMPUT_1:def 17</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="123_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:123_1_1_2_1"/>
<font color="Maroon" title="c2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">R</font> where <font color="Olive" title="b1">R</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> : <font color="Olive" title="b1">R</font> is <a href="comput_1.html#V7" title="COMPUT_1:attr.7">primitive-recursively_closed</a> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c3">R</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span><b> such that </b><br/><a NAME="E3:123_1_1_2_1"/><i><font color="Green" title="E86">A4</font></i>: 
( <font color="Maroon" title="c3">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c3">R</font> is <a href="comput_1.html#V7" title="COMPUT_1:attr.7">primitive-recursively_closed</a> )
 ;<br/><b>thus </b><a NAME="E4:123_1_1_2_1"/>
1 <a href="comput_1.html#K6" title="COMPUT_1:func.6">succ</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:123_1_1_2_1"><i><font color="Green" title="E86">A4</font></i></a>, <a class="ref" href="comput_1.html#D17" target="_self" title="COMPUT_1:def.17">Def17</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:123_1_1_2"/>
1 <a href="comput_1.html#K6" title="COMPUT_1:func.6">succ</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K10" title="COMPUT_1:func.10">PrimRec</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E2:123_1_1"><i><font color="Green" title="E84">A1</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c2">n</font>, <font color="Maroon" title="c3">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:123_1_1_2_2"/><i><font color="Green" title="E86">A5</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="123_1_1_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:123_1_1_2_2_1"/>
<font color="Maroon" title="c4">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">R</font> where <font color="Olive" title="b1">R</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> : <font color="Olive" title="b1">R</font> is <a href="comput_1.html#V7" title="COMPUT_1:attr.7">primitive-recursively_closed</a> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c5">R</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span><b> such that </b><br/><a NAME="E3:123_1_1_2_2_1"/><i><font color="Green" title="E87">A6</font></i>: 
( <font color="Maroon" title="c5">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Maroon" title="c5">R</font> is <a href="comput_1.html#V7" title="COMPUT_1:attr.7">primitive-recursively_closed</a> )
 ;<br/><b>thus </b><a NAME="E4:123_1_1_2_2_1"/>
<font color="Maroon" title="c2">n</font> <a href="comput_1.html#K7" title="COMPUT_1:func.7">proj</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:123_1_1_2_2"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="comput_1.html#E3:123_1_1_2_2_1"><i><font color="Green" title="E87">A6</font></i></a>, <a class="ref" href="comput_1.html#D17" target="_self" title="COMPUT_1:def.17">Def17</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:123_1_1_2_2"/>
<font color="Maroon" title="c2">n</font> <a href="comput_1.html#K7" title="COMPUT_1:func.7">proj</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K10" title="COMPUT_1:func.10">PrimRec</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E2:123_1_1"><i><font color="Green" title="E84">A1</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">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" title="COMPUT_1:def.15">COMPUT_1:def 15</a>
<div class="add"><b>let </b><font color="Maroon" title="c2">f</font> be    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c3">F</font> be  <a href="comput_1.html#V4" title="COMPUT_1:attr.4">with_the_same_arity</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:123_1_1_2_3"/><i><font color="Green" title="E86">A7</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K10" title="COMPUT_1:func.10">PrimRec</a>  &amp;  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="comput_1.html#K10" title="COMPUT_1:func.10">PrimRec</a>  )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="123_1_1_2_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:123_1_1_2_3_1"/><i><font color="Green" title="E87">A8</font></i>: 
<font color="Maroon" title="c4">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">R</font> where <font color="Olive" title="b1">R</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> : <font color="Olive" title="b1">R</font> is <a href="comput_1.html#V7" title="COMPUT_1:attr.7">primitive-recursively_closed</a> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c5">R</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span><b> such that </b><br/><a NAME="E3:123_1_1_2_3_1"/><i><font color="Green" title="E88">A9</font></i>: 
( <font color="Maroon" title="c5">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Y</font> &amp; <font color="Maroon" title="c5">R</font> is <a href="comput_1.html#V7" title="COMPUT_1:attr.7">primitive-recursively_closed</a> )
 ;<br/><a NAME="E4:123_1_1_2_3_1"/><i><font color="Green" title="E89">A10</font></i>: 
<font color="Maroon" title="c5">R</font> is <a href="comput_1.html#V5" title="COMPUT_1:attr.5">composition_closed</a>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:123_1_1_2_3_1"><i><font color="Green" title="E88">A9</font></i></a>, <a class="ref" href="comput_1.html#D17" target="_self" title="COMPUT_1:def.17">Def17</a></i>;<br/><a NAME="E5:123_1_1_2_3_1"/>
 <a href="comput_1.html#K10" title="COMPUT_1:func.10">PrimRec</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">R</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:123_1_1_2_3_1"><i><font color="Green" title="E87">A8</font></i></a>, <a class="txt" href="comput_1.html#E3:123_1_1_2_3_1"><i><font color="Green" title="E88">A9</font></i></a>, <a class="ref" href="setfam_1.html#T4" title="SETFAM_1:th.4">SETFAM_1:4</a></i>;<br/><a NAME="E6:123_1_1_2_3_1"/><b>then </b>
( <font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">R</font> )
 <b>by </b><i><a class="txt" href="comput_1.html#E1:123_1_1_2_3"><i><font color="Green" title="E86">A7</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E7:123_1_1_2_3_1"/>
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c3">F</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:123_1_1_2_3"><i><font color="Green" title="E86">A7</font></i></a>, <a class="txt" href="comput_1.html#E3:123_1_1_2_3_1"><i><font color="Green" title="E88">A9</font></i></a>, <a class="txt" href="comput_1.html#E4:123_1_1_2_3_1"><i><font color="Green" title="E89">A10</font></i></a>, <a class="ref" href="comput_1.html#D15" target="_self" title="COMPUT_1:def.15">Def15</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:123_1_1_2_3"/>
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c3">F</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K10" title="COMPUT_1:func.10">PrimRec</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E2:123_1_1"><i><font color="Green" title="E84">A1</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">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" title="COMPUT_1:def.16">COMPUT_1:def 16</a>
<div class="add"><b>let </b><font color="Maroon" title="c2">g</font>, <font color="Maroon" title="c3">f1</font>, <font color="Maroon" title="c4">f2</font> be    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:123_1_1_2_4"/><i><font color="Green" title="E86">A11</font></i>: 
<font color="Maroon" title="c2">g</font> <a href="comput_1.html#R1" title="COMPUT_1:pred.1">is_primitive-recursively_expressed_by</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c5">i</font>
 <b>and </b><br/><a NAME="E2:123_1_1_2_4"/><i><font color="Green" title="E87">A12</font></i>: 
<font color="Maroon" title="c3">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K10" title="COMPUT_1:func.10">PrimRec</a> 
 <b>and </b><br/><a NAME="E3:123_1_1_2_4"/><i><font color="Green" title="E88">A13</font></i>: 
<font color="Maroon" title="c4">f2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K10" title="COMPUT_1:func.10">PrimRec</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="123_1_1_2_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:123_1_1_2_4_1"/><i><font color="Green" title="E89">A14</font></i>: 
<font color="Maroon" title="c6">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">R</font> where <font color="Olive" title="b1">R</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> : <font color="Olive" title="b1">R</font> is <a href="comput_1.html#V7" title="COMPUT_1:attr.7">primitive-recursively_closed</a> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c7">R</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span><b> such that </b><br/><a NAME="E3:123_1_1_2_4_1"/><i><font color="Green" title="E90">A15</font></i>: 
( <font color="Maroon" title="c7">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Y</font> &amp; <font color="Maroon" title="c7">R</font> is <a href="comput_1.html#V7" title="COMPUT_1:attr.7">primitive-recursively_closed</a> )
 ;<br/><a NAME="E4:123_1_1_2_4_1"/><i><font color="Green" title="E91">A16</font></i>: 
<font color="Maroon" title="c7">R</font> is <a href="comput_1.html#V6" title="COMPUT_1:attr.6">primitive-recursion_closed</a>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:123_1_1_2_4_1"><i><font color="Green" title="E90">A15</font></i></a>, <a class="ref" href="comput_1.html#D17" target="_self" title="COMPUT_1:def.17">Def17</a></i>;<br/><a NAME="E5:123_1_1_2_4_1"/>
( <font color="Maroon" title="c3">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">R</font> &amp; <font color="Maroon" title="c4">f2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">R</font> )
 <b>by </b><i><a class="txt" href="comput_1.html#E2:123_1_1_2_4"><i><font color="Green" title="E87">A12</font></i></a>, <a class="txt" href="comput_1.html#E3:123_1_1_2_4"><i><font color="Green" title="E88">A13</font></i></a>, <a class="txt" href="comput_1.html#E1:123_1_1_2_4_1"><i><font color="Green" title="E89">A14</font></i></a>, <a class="txt" href="comput_1.html#E3:123_1_1_2_4_1"><i><font color="Green" title="E90">A15</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/><b>hence </b><a NAME="E6:123_1_1_2_4_1"/>
<font color="Maroon" title="c2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:123_1_1_2_4"><i><font color="Green" title="E86">A11</font></i></a>, <a class="txt" href="comput_1.html#E3:123_1_1_2_4_1"><i><font color="Green" title="E90">A15</font></i></a>, <a class="txt" href="comput_1.html#E4:123_1_1_2_4_1"><i><font color="Green" title="E91">A16</font></i></a>, <a class="ref" href="comput_1.html#D16" target="_self" title="COMPUT_1:def.16">Def16</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:123_1_1_2_4"/>
<font color="Maroon" title="c2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K10" title="COMPUT_1:func.10">PrimRec</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E2:123_1_1"><i><font color="Green" title="E84">A1</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/></div>
<b>end;</b></div>


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


</div>
