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

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

<b>set </b><font color="Maroon">prd</font> =  <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">f</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>   st <font color="Olive">f</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">a<sub>1</sub></font> holds <br/><font color="Olive">f</font> is <a href="unialg_1.html#V2">quasi_total</a>;<br/>
<a NAME="E1:137"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="137_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be    <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:137_1"/>
<font color="Maroon">f</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="E2:137_1"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="comput_1.html#K12" target="_self">initial-funcs</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:137_1"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">f</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">f</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) </span>}</span>  )
 
<b>by </b><i><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="137_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:137_1_1"/>
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 0 <a href="comput_1.html#K6" target="_self">const</a> 0 or <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 1 <a href="comput_1.html#K7" target="_self">succ</a> 1 or <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) </span>}</span>  )
 </span><b>by </b><i><a class="ref" href="comput_1.html#T19" target="_self">Th19</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="137_1_1_1"><b>suppose </b></a><a NAME="E1:137_1_1_1"/>
<font color="Maroon">f</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="E2:137_1_1_1"/>
<font color="Maroon">f</font> is <a href="unialg_1.html#V2">quasi_total</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="137_1_1_2"><b>suppose </b></a><a NAME="E1:137_1_1_2"/>
<font color="Maroon">f</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="E2:137_1_1_2"/>
<font color="Maroon">f</font> is <a href="unialg_1.html#V2">quasi_total</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="137_1_1_3"><b>suppose </b></a><a NAME="E1:137_1_1_3"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:137_1_1_3"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Maroon">i</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 ;<br/><b>thus </b><a NAME="E4:137_1_1_3"/>
<font color="Maroon">f</font> is <a href="unialg_1.html#V2">quasi_total</a>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:137"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">m</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="137_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:137_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">m</font>]
 ;<br/>

<b>let </b><font color="Maroon">f</font> be    <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E2:137_2"/>
<font color="Maroon">f</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">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E3:137_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</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">m</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">m</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="137_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:137_2_1"/>
( <font color="Maroon">f</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">m</font></span>)</span> or <font color="Maroon">f</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">m</font></span>)</span> )
 </span><b>by </b><i>, <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="137_2_1_1"><b>suppose </b></a><a NAME="E1:137_2_1_1"/>
<font color="Maroon">f</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">m</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:137_2_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">f</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">m</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> 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">f1</font>, <font color="Olive">f2</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">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">f1</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">m</font> &amp; <font color="Olive">f2</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">m</font> &amp; <font color="Olive">g</font> <a href="comput_1.html#R1" target="_self">is_primitive-recursively_expressed_by</a> <font color="Olive">f1</font>,<font color="Olive">f2</font>,<font color="Olive">i</font> ) </span>}</span> 
 ;<br/><b>thus </b><a NAME="E3:137_2_1_1"/>
<font color="Maroon">f</font> is <a href="unialg_1.html#V2">quasi_total</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="137_2_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="137_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="E1:137_2_1_1_1_1"/>
( <font color="Maroon">f</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">m</font> or <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> 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">f1</font>, <font color="Olive">f2</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">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">f1</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">m</font> &amp; <font color="Olive">f2</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">m</font> &amp; <font color="Olive">g</font> <a href="comput_1.html#R1" target="_self">is_primitive-recursively_expressed_by</a> <font color="Olive">f1</font>,<font color="Olive">f2</font>,<font color="Olive">i</font> ) </span>}</span>  )
 </span><b>by </b><i><a class="ref" href="comput_1.html#T20" target="_self">Th20</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="137_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:137_2_1_1_1_1_1"/>
<font color="Maroon">f</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">m</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:137_2_1_1_1_1_1"/>
<font color="Maroon">f</font> is <a href="unialg_1.html#V2">quasi_total</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="137_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:137_2_1_1_1_1_2"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> 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">f1</font>, <font color="Olive">f2</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">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">f1</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">m</font> &amp; <font color="Olive">f2</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">m</font> &amp; <font color="Olive">g</font> <a href="comput_1.html#R1" target="_self">is_primitive-recursively_expressed_by</a> <font color="Olive">f1</font>,<font color="Olive">f2</font>,<font color="Olive">i</font> ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">g</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="E3:137_2_1_1_1_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E4:137_2_1_1_1_1_2"/><i><font color="Green">E58</font></i>: 
 ex <font color="Olive">f1</font>, <font color="Olive">f2</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">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">f1</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">m</font> &amp; <font color="Olive">f2</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">m</font> &amp; <font color="Maroon">g</font> <a href="comput_1.html#R1" target="_self">is_primitive-recursively_expressed_by</a> <font color="Olive">f1</font>,<font color="Olive">f2</font>,<font color="Olive">i</font> )
 ;<br/><b>consider </b><font color="Maroon">f1</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">f2</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">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:137_2_1_1_1_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">f1</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">m</font> &amp; <font color="Maroon">f2</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">m</font> )
 <b>and </b><br/><a NAME="E7:137_2_1_1_1_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">g</font> <a href="comput_1.html#R1" target="_self">is_primitive-recursively_expressed_by</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="comput_1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E8:137_2_1_1_1_1_2"/>
( <font color="Maroon">f1</font> is <a href="unialg_1.html#V2">quasi_total</a> &amp; <font color="Maroon">f2</font> is <a href="unialg_1.html#V2">quasi_total</a> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E9:137_2_1_1_1_1_2"/>
<font color="Maroon">f</font> is <a href="unialg_1.html#V2">quasi_total</a>
 <b>by </b><i>, , </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="137_2_1_2"><b>suppose </b></a><a NAME="E1:137_2_1_2"/>
<font color="Maroon">f</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">m</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:137_2_1_2"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">f</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">m</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">h</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">F</font></span><a href="funct_6.html#K6">:&gt;</a></span></span>)</span> where <font color="Olive">h</font> 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> , <font color="Olive">F</font> 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">h</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">m</font> &amp;  <a href="comput_1.html#K3" target="_self">arity</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">F</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">F</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">m</font> ) </span>}</span> 
 ;<br/><b>thus </b><a NAME="E3:137_2_1_2"/>
<font color="Maroon">f</font> is <a href="unialg_1.html#V2">quasi_total</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="137_2_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="137_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="E1:137_2_1_2_1_1"/>
( <font color="Maroon">f</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">m</font> or <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">h</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">F</font></span><a href="funct_6.html#K6">:&gt;</a></span></span>)</span> where <font color="Olive">h</font> 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> , <font color="Olive">F</font> 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">h</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">m</font> &amp;  <a href="comput_1.html#K3" target="_self">arity</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">F</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">F</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">m</font> ) </span>}</span>  )
 </span><b>by </b><i><a class="ref" href="comput_1.html#T20" target="_self">Th20</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="137_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:137_2_1_2_1_1_1"/>
<font color="Maroon">f</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">m</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:137_2_1_2_1_1_1"/>
<font color="Maroon">f</font> is <a href="unialg_1.html#V2">quasi_total</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="137_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:137_2_1_2_1_1_2"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">h</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">F</font></span><a href="funct_6.html#K6">:&gt;</a></span></span>)</span> where <font color="Olive">h</font> 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> , <font color="Olive">F</font> 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">h</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">m</font> &amp;  <a href="comput_1.html#K3" target="_self">arity</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">F</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">F</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">m</font> ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">h</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">F</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="E3:137_2_1_2_1_1_2"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">F</font></span><a href="funct_6.html#K6">:&gt;</a></span> &amp; <font color="Maroon">h</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">m</font> &amp;  <a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <a href="comput_1.html#K15" target="_self">PrimRec-Approximation</a>  <a href="prob_1.html#K1">.</a> <font color="Maroon">m</font> )
 ;<br/><a NAME="E4:137_2_1_2_1_1_2"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">h</font> is <a href="unialg_1.html#V2">quasi_total</a>
 <b>by </b><i>, </i>;<br/><a NAME="E5:137_2_1_2_1_1_2"/>
for <font color="Olive">h</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>   st <font color="Olive">h</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> holds <br/><font color="Olive">h</font> is <a href="unialg_1.html#V2">quasi_total</a>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E6:137_2_1_2_1_1_2"/>
<font color="Maroon">f</font> is <a href="unialg_1.html#V2">quasi_total</a>
 <b>by </b><i>, <a class="ref" href="comput_1.html#D5" target="_self">Def5</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="E3:137"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">m</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="comput_1.html#D4" target="_self">Def4</a>, <a class="ref" href="comput_1.html#T19" target="_self">Th19</a>);<br/></i>
<b>hence </b><a NAME="E4:137"/>
for <font color="Olive">f</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>   st <font color="Olive">f</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">m</font> holds <br/><font color="Olive">f</font> is <a href="unialg_1.html#V2">quasi_total</a>
 ;<br/>


</div>
