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

<b>set </b><font color="Maroon">c<sub>1</sub></font> =  <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a> ;<br/>
<a NAME="E1:136"/><i><font color="Green">E92</font></i>: 
 <a href="comput_1.html#K11" target="_self">PrimRec</a>  <a href="tarski.html#R1">c=</a>  <a href="prob_1.html#K2">Union</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a> 
 
<b>by </b><i><a class="ref" href="comput_1.html#T78" target="_self">Th78</a>, <a class="ref" href="comput_1.html#T80" target="_self">Th80</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="comput_1.html#K11" target="_self">PrimRec</a> ;<br/>
<a NAME="E2:136"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></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="E1:136_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> 0
 ;<br/>

<a NAME="E2:136_1"/>
<a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default">0 <a href="comput_1.html#K6" target="_self">const</a> 0</span>)</span>,<span class="p2">(<span class="default">1 <a href="comput_1.html#K7" target="_self">succ</a> 1</span>)</span></span><a href="tarski.html#K2">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a>, B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="comput_1.html#D23" target="_self">Def23</a></i>;<br/>
<a NAME="E3:136_1"/><b>then </b><i><font color="Green">E95</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default">0 <a href="comput_1.html#K6" target="_self">const</a> 0</span>)</span>,<span class="p2">(<span class="default">1 <a href="comput_1.html#K7" target="_self">succ</a> 1</span>)</span></span><a href="tarski.html#K2">}</a></span> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a>, B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  )
 
<b>by </b><i><a class="txt" href="comput_1.html#E1:136_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:136_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="comput_1.html#K6" target="_self">const</a> 0 or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="comput_1.html#K7" target="_self">succ</a> 1 or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a>, B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="comput_1.html#E3:136_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_1_1_1"><b>suppose </b></a><a NAME="E1:136_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="comput_1.html#K6" target="_self">const</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:136_1_1_1"/>
<font color="Maroon">c<sub>2</sub></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#D17" target="_self">Def17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_1_1_2"><b>suppose </b></a><a NAME="E1:136_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="comput_1.html#K7" target="_self">succ</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:136_1_1_2"/>
<font color="Maroon">c<sub>2</sub></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#D17" target="_self">Def17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_1_1_3"><b>suppose </b></a><a NAME="E1:136_1_1_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a>, B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:136_1_1_3"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Maroon">c<sub>4</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><b>thus </b><a NAME="E4:136_1_1_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="comput_1.html#K11" target="_self">PrimRec</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E3:136_1_1_3"><i><font color="Green">E96</font></i></a>, <a class="ref" href="comput_1.html#D17" target="_self">Def17</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<div><i><font color="Green">E94</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="136_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:136_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 ;<br/><b>thus </b><a NAME="E2:136_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></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="E1:136_2_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E2:136_2_1"/><i><font color="Green">E97</font></i>: 
<a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K13" target="_self">PR-closure</a> <span class="p2">(<span class="default"><a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="comput_1.html#K14" target="_self">composition-closure</a> <span class="p2">(<span class="default"><a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="comput_1.html#D23" target="_self">Def23</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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>  : ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <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> ex <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="comput_1.html#R1" target="_self">is_primitive-recursively_expressed_by</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="funct_6.html#K6">:&gt;</a></span></span>)</span> where B 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 is  <a href="comput_1.html#V5" target="_self">with_the_same_arity</a>  <a href="finseq_2.html#M2">Element</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> <a href="finseq_2.html#K3">*</a>  : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="comput_1.html#K3" target="_self">arity</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:136_2_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="comput_1.html#K13" target="_self">PR-closure</a> <span class="p1">(<span class="default"><a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="comput_1.html#K14" target="_self">composition-closure</a> <span class="p1">(<span class="default"><a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="comput_1.html#E1:136_2_1"><i><font color="Green">E96</font></i></a>, <a class="txt" href="comput_1.html#E2:136_2_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_2_1_1_1"><b>suppose </b></a><a NAME="E1:136_2_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="comput_1.html#K13" target="_self">PR-closure</a> <span class="p1">(<span class="default"><a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:136_2_1_1_1"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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>  : ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <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> ex <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="comput_1.html#R1" target="_self">is_primitive-recursively_expressed_by</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> ) </span>}</span> 
 ;<br/><b>thus </b><a NAME="E3:136_2_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="comput_1.html#K11" target="_self">PrimRec</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_2_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:136_2_1_1_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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>  : ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <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> ex <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="comput_1.html#R1" target="_self">is_primitive-recursively_expressed_by</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> ) </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="comput_1.html#E2:136_2_1_1_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:136_2_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:136_2_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="comput_1.html#K11" target="_self">PrimRec</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E1:136_2"><i><font color="Green">E95</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:136_2_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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>  : ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <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> ex <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="comput_1.html#R1" target="_self">is_primitive-recursively_expressed_by</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <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> such that </b><br/><a NAME="E3:136_2_1_1_1_1_1_2"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <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> ex <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="comput_1.html#R1" target="_self">is_primitive-recursively_expressed_by</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> ) )
 ;<br/><a NAME="E4:136_2_1_1_1_1_1_2"/>
 <a href="comput_1.html#K11" target="_self">PrimRec</a>  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/><b>hence </b><a NAME="E5:136_2_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="comput_1.html#K11" target="_self">PrimRec</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E1:136_2"><i><font color="Green">E95</font></i></a>, <a class="txt" href="comput_1.html#E3:136_2_1_1_1_1_1_2"><i><font color="Green">E99</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>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_2_1_1_2"><b>suppose </b></a><a NAME="E1:136_2_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="comput_1.html#K14" target="_self">composition-closure</a> <span class="p1">(<span class="default"><a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:136_2_1_1_2"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="funct_6.html#K6">:&gt;</a></span></span>)</span> where B 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 is  <a href="comput_1.html#V5" target="_self">with_the_same_arity</a>  <a href="finseq_2.html#M2">Element</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> <a href="finseq_2.html#K3">*</a>  : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="comput_1.html#K3" target="_self">arity</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
 ;<br/><b>thus </b><a NAME="E3:136_2_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="comput_1.html#K11" target="_self">PrimRec</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_2_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:136_2_1_1_2_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="funct_6.html#K6">:&gt;</a></span></span>)</span> where B 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 is  <a href="comput_1.html#V5" target="_self">with_the_same_arity</a>  <a href="finseq_2.html#M2">Element</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> <a href="finseq_2.html#K3">*</a>  : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="comput_1.html#K3" target="_self">arity</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="comput_1.html#E2:136_2_1_1_2"><i><font color="Green">E98</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:136_2_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:136_2_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="comput_1.html#K11" target="_self">PrimRec</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E1:136_2"><i><font color="Green">E95</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:136_2_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="funct_6.html#K6">:&gt;</a></span></span>)</span> where B 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 is  <a href="comput_1.html#V5" target="_self">with_the_same_arity</a>  <a href="finseq_2.html#M2">Element</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> <a href="finseq_2.html#K3">*</a>  : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="comput_1.html#K3" target="_self">arity</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <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">c<sub>7</sub></font> being  <a href="comput_1.html#V5" target="_self">with_the_same_arity</a>  <a href="finseq_2.html#M2">Element</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> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E3:136_2_1_1_2_1_1_2"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</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> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><a NAME="E4:136_2_1_1_2_1_1_2"/><i><font color="Green">E100</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="comput_1.html#K11" target="_self">PrimRec</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E1:136_2"><i><font color="Green">E95</font></i></a>, <a class="txt" href="comput_1.html#E3:136_2_1_1_2_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E5:136_2_1_1_2_1_1_2"/>
 <a href="comput_1.html#K11" target="_self">PrimRec</a>  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>hence </b><a NAME="E6:136_2_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="comput_1.html#K11" target="_self">PrimRec</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E1:136_2"><i><font color="Green">E95</font></i></a>, <a class="txt" href="comput_1.html#E3:136_2_1_1_2_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="comput_1.html#E4:136_2_1_1_2_1_1_2"><i><font color="Green">E100</font></i></a>, <a class="ref" href="comput_1.html#D15" target="_self">Def15</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E4:136"/><i><font color="Green">E95</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#E2:136"><i><font color="Green">E93</font></i></a>, <a class="txt" href="comput_1.html#E3:136"><i><font color="Green">E94</font></i></a>);<br/></i>
<a NAME="E5:136"/>
 <a href="prob_1.html#K2">Union</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="tarski.html#R1">c=</a>  <a href="comput_1.html#K11" target="_self">PrimRec</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></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="E1:136_3"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K2">Union</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="comput_1.html#K11" target="_self">PrimRec</a>  )
 ;<br/>

<a NAME="E2:136_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:136_3"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:136_3"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  &amp; <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="comput_1.html#E4:136_3"><i><font color="Green">E97</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="comput_1.html#E6:136_3"><i><font color="Green">E98</font></i></a></i>;<br/>
<a NAME="E8:136_3"/>
<a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="comput_1.html#K11" target="_self">PrimRec</a> 
 
<b>by </b><i><a class="txt" href="comput_1.html#E4:136"><i><font color="Green">E95</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:136_3"/>
contradiction
 <b>by </b><i><a class="txt" href="comput_1.html#E1:136_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="comput_1.html#E4:136_3"><i><font color="Green">E97</font></i></a>, <a class="txt" href="comput_1.html#E6:136_3"><i><font color="Green">E98</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:136"/>
 <a href="comput_1.html#K11" target="_self">PrimRec</a>  <a href="hidden.html#R1">=</a>  <a href="prob_1.html#K2">Union</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E1:136"><i><font color="Green">E92</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
