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

<b>let </b><font color="Maroon" title="c1">m</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>set </b><font color="Maroon" title="c2">prd</font> =  <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <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> ] <b>means </b>for <font color="Olive" title="b1">f</font> being   <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>   st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> $1 holds <br/><font color="Olive" title="b1">f</font> is <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>;<br/>
<a NAME="E1:135"/><i><font color="Green" title="E90">A1</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="135_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">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>assume </b><a NAME="E1:135_1"/>
<font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> 0
 ;<br/>

<a NAME="E2:135_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K11" title="COMPUT_1:func.11">initial-funcs</a> 
 
<b>by </b><i><a class="ref" href="comput_1.html#D23" target="_self" title="COMPUT_1:def.23">Def23</a></i>;<br/>
<a NAME="E3:135_1"/><b>then </b><i><font color="Green" title="E91">A2</font></i>: 
( <font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default">0 <a href="comput_1.html#K5" title="COMPUT_1:func.5">const</a> 0</span>)</span>,<span class="p2">(<span class="default">1 <a href="comput_1.html#K6" title="COMPUT_1:func.6">succ</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="comput_1.html#K7" title="COMPUT_1:func.7">proj</a> <font color="Olive" title="b2">i</font></span>)</span> where <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">i</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) </span>}</span>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="135_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:135_1_1"/>
( <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 <a href="comput_1.html#K5" title="COMPUT_1:func.5">const</a> 0 or <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="comput_1.html#K6" title="COMPUT_1:func.6">succ</a> 1 or <font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="comput_1.html#K7" title="COMPUT_1:func.7">proj</a> <font color="Olive" title="b2">i</font></span>)</span> where <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">i</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="comput_1.html#E3:135_1"><i><font color="Green" title="E91">A2</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="135_1_1_1"><b>suppose </b></a><a NAME="E1:135_1_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 <a href="comput_1.html#K5" title="COMPUT_1:func.5">const</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:135_1_1_1"/>
<font color="Maroon" title="c3">f</font> is <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>
 <b>by </b><i><a class="ref" href="comput_1.html#T32" target="_self" title="COMPUT_1:th.32">Th32</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="135_1_1_2"><b>suppose </b></a><a NAME="E1:135_1_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="comput_1.html#K6" title="COMPUT_1:func.6">succ</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:135_1_1_2"/>
<font color="Maroon" title="c3">f</font> is <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>
 <b>by </b><i><a class="ref" href="comput_1.html#T32" target="_self" title="COMPUT_1:th.32">Th32</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="135_1_1_3"><b>suppose </b></a><a NAME="E1:135_1_1_3"/>
<font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="comput_1.html#K7" title="COMPUT_1:func.7">proj</a> <font color="Olive" title="b2">i</font></span>)</span> where <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">i</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c4">n</font>, <font color="Maroon" title="c5">i</font> being    <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> <b> such that </b><br/><a NAME="E3:135_1_1_3"/><i><font color="Green" title="E92">A3</font></i>: 
( <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="comput_1.html#K7" title="COMPUT_1:func.7">proj</a> <font color="Maroon" title="c5">i</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> )
 ;<br/><b>thus </b><a NAME="E4:135_1_1_3"/>
<font color="Maroon" title="c3">f</font> is <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:135_1_1_3"><i><font color="Green" title="E92">A3</font></i></a>, <a class="ref" href="comput_1.html#T32" target="_self" title="COMPUT_1:th.32">Th32</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:135"/><i><font color="Green" title="E91">A4</font></i>: 
for <font color="Olive" title="b1">m</font> being   <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>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">m</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="135_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">m</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="E1:135_2"/><i><font color="Green" title="E92">A5</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c3">m</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c4">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>assume </b><a NAME="E2:135_2"/>
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E3:135_2"/><b>then </b><i><font color="Green" title="E93">A6</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="comput_1.html#K12" title="COMPUT_1:func.12">PR-closure</a> <span class="p2">(<span class="default"><a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="comput_1.html#K13" title="COMPUT_1:func.13">composition-closure</a> <span class="p2">(<span class="default"><a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="comput_1.html#D23" target="_self" title="COMPUT_1:def.23">Def23</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="135_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:135_2_1"/>
( <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K12" title="COMPUT_1:func.12">PR-closure</a> <span class="p1">(<span class="default"><a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">m</font></span>)</span> or <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K13" title="COMPUT_1:func.13">composition-closure</a> <span class="p1">(<span class="default"><a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">m</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="comput_1.html#E3:135_2"><i><font color="Green" title="E93">A6</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="135_2_1_1"><b>suppose </b></a><a NAME="E1:135_2_1_1"/><i><font color="Green" title="E94">A7</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K12" title="COMPUT_1:func.12">PR-closure</a> <span class="p1">(<span class="default"><a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">m</font></span>)</span>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:135_2_1_1"/>
<font color="Maroon" title="c4">f</font> is <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="135_2_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="135_2_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="E1:135_2_1_1_1_1"/>
( <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">m</font> or <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is    <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>  :  ex <font color="Olive" title="b2">f1</font>, <font color="Olive" title="b3">f2</font> being   <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>  ex <font color="Olive" title="b4">i</font> being   <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>  st <br/>( <font color="Olive" title="b2">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Olive" title="b3">f2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Olive" title="b1">g</font> <a href="comput_1.html#R1" title="COMPUT_1:pred.1">is_primitive-recursively_expressed_by</a> <font color="Olive" title="b2">f1</font>,<font color="Olive" title="b3">f2</font>,<font color="Olive" title="b4">i</font> ) </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="comput_1.html#E1:135_2_1_1"><i><font color="Green" title="E94">A7</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="135_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:135_2_1_1_1_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">m</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:135_2_1_1_1_1_1"/>
<font color="Maroon" title="c4">f</font> is <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:135_2"><i><font color="Green" title="E92">A5</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="135_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:135_2_1_1_1_1_2"/>
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is    <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>  :  ex <font color="Olive" title="b2">f1</font>, <font color="Olive" title="b3">f2</font> being   <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>  ex <font color="Olive" title="b4">i</font> being   <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>  st <br/>( <font color="Olive" title="b2">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Olive" title="b3">f2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Olive" title="b1">g</font> <a href="comput_1.html#R1" title="COMPUT_1:pred.1">is_primitive-recursively_expressed_by</a> <font color="Olive" title="b2">f1</font>,<font color="Olive" title="b3">f2</font>,<font color="Olive" title="b4">i</font> ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">g</font> being    <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> <b> such that </b><br/><a NAME="E3:135_2_1_1_1_1_2"/><i><font color="Green" title="E95">A8</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font>
 <b>and </b><br/><a NAME="E4:135_2_1_1_1_1_2"/><i><font color="Green" title="E96">A9</font></i>: 
 ex <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being   <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>  ex <font color="Olive" title="b3">i</font> being   <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>  st <br/>( <font color="Olive" title="b1">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Olive" title="b2">f2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c5">g</font> <a href="comput_1.html#R1" title="COMPUT_1:pred.1">is_primitive-recursively_expressed_by</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font>,<font color="Olive" title="b3">i</font> )
 ;<br/><b>consider </b><font color="Maroon" title="c6">f1</font>, <font color="Maroon" title="c7">f2</font> being    <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> , <font color="Maroon" title="c8">i</font> being    <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> <b> such that </b><br/><a NAME="E6:135_2_1_1_1_1_2"/><i><font color="Green" title="E97">A10</font></i>: 
( <font color="Maroon" title="c6">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c7">f2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">m</font> )
 <b>and </b><br/><a NAME="E7:135_2_1_1_1_1_2"/><i><font color="Green" title="E98">A11</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="comput_1.html#R1" title="COMPUT_1:pred.1">is_primitive-recursively_expressed_by</a> <font color="Maroon" title="c6">f1</font>,<font color="Maroon" title="c7">f2</font>,<font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E4:135_2_1_1_1_1_2"><i><font color="Green" title="E96">A9</font></i></a></i>;<br/><a NAME="E8:135_2_1_1_1_1_2"/>
( <font color="Maroon" title="c6">f1</font> is <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a> &amp; <font color="Maroon" title="c7">f2</font> is <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a> )
 <b>by </b><i><a class="txt" href="comput_1.html#E1:135_2"><i><font color="Green" title="E92">A5</font></i></a>, <a class="txt" href="comput_1.html#E6:135_2_1_1_1_1_2"><i><font color="Green" title="E97">A10</font></i></a></i>;<br/><b>hence </b><a NAME="E9:135_2_1_1_1_1_2"/>
<font color="Maroon" title="c4">f</font> is <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:135_2_1_1_1_1_2"><i><font color="Green" title="E95">A8</font></i></a>, <a class="txt" href="comput_1.html#E7:135_2_1_1_1_1_2"><i><font color="Green" title="E98">A11</font></i></a>, <a class="ref" href="comput_1.html#T72" target="_self" title="COMPUT_1:th.72">Th72</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="135_2_1_2"><b>suppose </b></a><a NAME="E1:135_2_1_2"/><i><font color="Green" title="E94">A12</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K13" title="COMPUT_1:func.13">composition-closure</a> <span class="p1">(<span class="default"><a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">m</font></span>)</span>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:135_2_1_2"/>
<font color="Maroon" title="c4">f</font> is <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="135_2_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="135_2_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="E1:135_2_1_2_1_1"/>
( <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">m</font> or <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Olive" title="b2">F</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span></span>)</span> where <font color="Olive" title="b1">h</font> is    <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> , <font color="Olive" title="b2">F</font> is  <a href="comput_1.html#V4" title="COMPUT_1:attr.4">with_the_same_arity</a>  <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</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> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">m</font> &amp;  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Olive" title="b1">h</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="Olive" title="b2">F</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">m</font> ) </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="comput_1.html#E1:135_2_1_2"><i><font color="Green" title="E94">A12</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="135_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:135_2_1_2_1_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">m</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:135_2_1_2_1_1_1"/>
<font color="Maroon" title="c4">f</font> is <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:135_2"><i><font color="Green" title="E92">A5</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="135_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:135_2_1_2_1_1_2"/>
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Olive" title="b2">F</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span></span>)</span> where <font color="Olive" title="b1">h</font> is    <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> , <font color="Olive" title="b2">F</font> is  <a href="comput_1.html#V4" title="COMPUT_1:attr.4">with_the_same_arity</a>  <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</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> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  : ( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">m</font> &amp;  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Olive" title="b1">h</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="Olive" title="b2">F</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">m</font> ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">h</font> being    <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> , <font color="Maroon" title="c6">F</font> being  <a href="comput_1.html#V4" title="COMPUT_1:attr.4">with_the_same_arity</a>  <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</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> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <b> such that </b><br/><a NAME="E3:135_2_1_2_1_1_2"/><i><font color="Green" title="E95">A13</font></i>: 
( <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</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="c6">F</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> &amp; <font color="Maroon" title="c5">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">m</font> &amp;  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c5">h</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="c6">F</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c3">m</font> )
 ;<br/><a NAME="E4:135_2_1_2_1_1_2"/><i><font color="Green" title="E96">A14</font></i>: 
<font color="Maroon" title="c5">h</font> is <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:135_2"><i><font color="Green" title="E92">A5</font></i></a>, <a class="txt" href="comput_1.html#E3:135_2_1_2_1_1_2"><i><font color="Green" title="E95">A13</font></i></a></i>;<br/><a NAME="E5:135_2_1_2_1_1_2"/>
for <font color="Olive" title="b1">h</font> being   <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>   st <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font> holds <br/><font color="Olive" title="b1">h</font> is <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:135_2"><i><font color="Green" title="E92">A5</font></i></a>, <a class="txt" href="comput_1.html#E3:135_2_1_2_1_1_2"><i><font color="Green" title="E95">A13</font></i></a></i>;<br/><b>hence </b><a NAME="E6:135_2_1_2_1_1_2"/>
<font color="Maroon" title="c4">f</font> is <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:135_2_1_2_1_1_2"><i><font color="Green" title="E95">A13</font></i></a>, <a class="txt" href="comput_1.html#E4:135_2_1_2_1_1_2"><i><font color="Green" title="E96">A14</font></i></a>, <a class="ref" href="comput_1.html#T50" target="_self" title="COMPUT_1:th.50">Th50</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>
<a NAME="E3:135"/>
for <font color="Olive" title="b1">m</font> being   <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>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">m</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="comput_1.html#E1:135"><i><font color="Green" title="E90">A1</font></i></a>, <a class="txt" href="comput_1.html#E2:135"><i><font color="Green" title="E91">A4</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:135"/>
for <font color="Olive" title="b1">f</font> being   <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>   st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="comput_1.html#K14" title="COMPUT_1:func.14">PrimRec-Approximation</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Olive" title="b1">f</font> is <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>
 ;<br/>


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