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

<b>let </b><font color="Maroon" title="c1">g</font> be  <a href="comput_1.html#V8" title="COMPUT_1:attr.8">primitive-recursive</a> <a href="comput_1.html#V11" title="COMPUT_1:attr.11">binary</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>

<a NAME="E1:155"/><i><font color="Green" title="E101">A1</font></i>: 
 <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="ref" href="comput_1.html#D26" target="_self" title="COMPUT_1:def.26">Def26</a></i>;<br/>
<a NAME="E2:155"/>
( 3 <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K9" title="COMPUT_1:func.9">PrimRec</a>  &amp; 3 <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> 3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K9" title="COMPUT_1:func.9">PrimRec</a>  )
 
<b>by </b><i><a class="ref" href="comput_1.html#D17" target="_self" title="COMPUT_1:def.17">Def17</a></i>;<br/>
<a NAME="E3:155"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default">3 <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> 3</span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="comput_1.html#K9" title="COMPUT_1:func.9">PrimRec</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T121" title="FINSEQ_2:th.121">FINSEQ_2:121</a></i>;<br/>
<a NAME="E4:155"/><b>then </b><i><font color="Green" title="E102">A2</font></i>: 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default">3 <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> 3</span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="comput_1.html#K9" title="COMPUT_1:func.9">PrimRec</a> 
 
;<br/>
<a NAME="E5:155"/>
2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="comput_1.html#K9" title="COMPUT_1:func.9">PrimRec</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="comput_1.html#K9" title="COMPUT_1:func.9">PrimRec</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<b>by </b><i><a class="ref" href="msualg_1.html#T12" title="MSUALG_1:th.12">MSUALG_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c2">F</font> = <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default">3 <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> 3</span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="comput_1.html#K9" title="COMPUT_1:func.9">PrimRec</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="txt" href="comput_1.html#E4:155"><i><font color="Green" title="E102">A2</font></i></a></i>;<br/>
<a NAME="E7:155"/>
<font color="Maroon" title="c2">F</font> is <a href="comput_1.html#V4" title="COMPUT_1:attr.4">with_the_same_arity</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="155_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="comput_1.html#D6" target="_self" title="COMPUT_1:def.6">COMPUT_1:def 6</a><br/>

<b>assume </b><a NAME="E1:155_1"/><i><font color="Green" title="E103">A3</font></i>: 
( <font color="Maroon" title="c3">f</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="c2">F</font> &amp; <font color="Maroon" title="c4">g</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="c2">F</font> )
 ;<br/>

<a NAME="E2:155_1"/><i><font color="Green" title="E104">A4</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default">3 <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> 3</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T147" title="FINSEQ_2:th.147">FINSEQ_2:147</a></i>;<br/>
<a NAME="E3:155_1"/><b>then </b><i><font color="Green" title="E105">A5</font></i>: 
( <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> 1 or <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> 3 )
 
<b>by </b><i><a class="txt" href="comput_1.html#E1:155_1"><i><font color="Green" title="E103">A3</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E4:155_1"/><i><font color="Green" title="E106">A6</font></i>: 
( <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> 1 or <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> 3 )
 
<b>by </b><i><a class="txt" href="comput_1.html#E1:155_1"><i><font color="Green" title="E103">A3</font></i></a>, <a class="txt" href="comput_1.html#E2:155_1"><i><font color="Green" title="E104">A4</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>thus </b><a NAME="E5:155_1"/>
(  not <font color="Maroon" title="c3">f</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or <font color="Maroon" title="c4">g</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="comput_1.html#E1:155_1"><i><font color="Green" title="E103">A3</font></i></a>, <a class="txt" href="comput_1.html#E2:155_1"><i><font color="Green" title="E104">A4</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>

<b>assume </b><a NAME="E6:155_1"/>
( not <font color="Maroon" title="c3">f</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c4">g</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 ;<br/>

<b>take </b>
3
;<br/>

<b>take </b>
 <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
;<br/>

<b>thus </b><a NAME="E7:155_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 3 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E3:155_1"><i><font color="Green" title="E105">A5</font></i></a>, <a class="ref" href="comput_1.html#T40" target="_self" title="COMPUT_1:th.40">Th40</a></i>;<br/>

<b>thus </b><a NAME="E8:155_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 3 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E4:155_1"><i><font color="Green" title="E106">A6</font></i></a>, <a class="ref" href="comput_1.html#T40" target="_self" title="COMPUT_1:th.40">Th40</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c3">F</font> = <font color="Maroon" title="c2">F</font> as  <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 <a href="comput_1.html#K9" title="COMPUT_1:func.9">PrimRec</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  ;<br/>
<a NAME="E9:155"/><i><font color="Green" title="E103">A7</font></i>: 
<font color="Maroon" title="c1">g</font> is    <a href="comput_1.html#M1" title="COMPUT_1:mode.1">Element</a> of  <a href="comput_1.html#K9" title="COMPUT_1:func.9">PrimRec</a> 
 
<b>by </b><i><a class="ref" href="comput_1.html#D19" target="_self" title="COMPUT_1:def.19">Def19</a></i>;<br/>
<a NAME="E10:155"/>
 <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F</font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E1:155"><i><font color="Green" title="E101">A1</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<b>hence </b><a NAME="E11:155"/>
 <a href="comput_1.html#K14" title="COMPUT_1:func.14">(1,2)-&gt;(1,?,2)</a> <font color="Maroon" title="c1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K9" title="COMPUT_1:func.9">PrimRec</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E9:155"><i><font color="Green" title="E103">A7</font></i></a>, <a class="ref" href="comput_1.html#T76" target="_self" title="COMPUT_1:th.76">Th76</a></i>;<br/>


</div>
