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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="comput_1.html#V9" target="_self">primitive-recursive</a> <a href="comput_1.html#V12" target="_self">binary</a> <a href="funct_1.html#NM1">Function</a>;<br/>

<a NAME="E1:158"/><i><font color="Green">E104</font></i>: 
 <a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="comput_1.html#D26" target="_self">Def26</a></i>;<br/>
<a NAME="E2:158"/>
( 3 <a href="comput_1.html#K8" target="_self">proj</a> 1 <a href="hidden.html#R2">in</a>  <a href="comput_1.html#K11" target="_self">PrimRec</a>  &amp; 3 <a href="comput_1.html#K8" target="_self">proj</a> 3 <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/>
<a NAME="E3:158"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default">3 <a href="comput_1.html#K8" target="_self">proj</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="comput_1.html#K8" target="_self">proj</a> 3</span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span> is    <a href="finseq_2.html#M2">Element</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <a href="comput_1.html#K11" target="_self">PrimRec</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T121">FINSEQ_2:121</a></i>;<br/>
<a NAME="E4:158"/><b>then </b><i><font color="Green">E105</font></i>: 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default">3 <a href="comput_1.html#K8" target="_self">proj</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="comput_1.html#K8" target="_self">proj</a> 3</span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R2">in</a> 2 <a href="finseq_2.html#K4">-tuples_on</a> <a href="comput_1.html#K11" target="_self">PrimRec</a> 
 
;<br/>
<a NAME="E5:158"/>
2 <a href="finseq_2.html#K4">-tuples_on</a> <a href="comput_1.html#K11" target="_self">PrimRec</a>  <a href="tarski.html#R1">c=</a> <a href="comput_1.html#K11" target="_self">PrimRec</a>  <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i><a class="ref" href="msualg_1.html#T12">MSUALG_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default">3 <a href="comput_1.html#K8" target="_self">proj</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="comput_1.html#K8" target="_self">proj</a> 3</span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of <a href="comput_1.html#K11" target="_self">PrimRec</a>  <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="txt" href="comput_1.html#E4:158"><i><font color="Green">E105</font></i></a></i>;<br/>
<a NAME="E7:158"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="comput_1.html#V5" target="_self">with_the_same_arity</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="158_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="funct_1.html#NM1">Function</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="comput_1.html#D6" target="_self">COMPUT_1:def 6</a><br/>

<b>assume </b><a NAME="E1:158_1"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E2:158_1"/><i><font color="Green">E107</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default">3 <a href="comput_1.html#K8" target="_self">proj</a> 1</span>)</span>,<span class="p2">(<span class="default">3 <a href="comput_1.html#K8" target="_self">proj</a> 3</span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T147">FINSEQ_2:147</a></i>;<br/>
<a NAME="E3:158_1"/><b>then </b><i><font color="Green">E108</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="comput_1.html#K8" target="_self">proj</a> 1 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="comput_1.html#K8" target="_self">proj</a> 3 )
 
<b>by </b><i><a class="txt" href="comput_1.html#E1:158_1"><i><font color="Green">E106</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E4:158_1"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="comput_1.html#K8" target="_self">proj</a> 1 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="comput_1.html#K8" target="_self">proj</a> 3 )
 
<b>by </b><i><a class="txt" href="comput_1.html#E1:158_1"><i><font color="Green">E106</font></i></a>, <a class="txt" href="comput_1.html#E2:158_1"><i><font color="Green">E107</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>thus </b><a NAME="E5:158_1"/>
(  not <font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">empty</a> or <font color="Maroon">c<sub>4</sub></font> is <a href="xboole_0.html#V1">empty</a> or  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="txt" href="comput_1.html#E1:158_1"><i><font color="Green">E106</font></i></a>, <a class="txt" href="comput_1.html#E2:158_1"><i><font color="Green">E107</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>

<b>assume </b><a NAME="E6:158_1"/>
( not <font color="Maroon">c<sub>3</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; not <font color="Maroon">c<sub>4</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 ;<br/>

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

<b>take </b>
 <a href="numbers.html#K5">NAT</a> 
;<br/>

<b>thus </b><a NAME="E7:158_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> 3 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E3:158_1"><i><font color="Green">E108</font></i></a>, <a class="ref" href="comput_1.html#T40" target="_self">Th40</a></i>;<br/>

<b>thus </b><a NAME="E8:158_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> 3 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E4:158_1"><i><font color="Green">E109</font></i></a>, <a class="ref" href="comput_1.html#T40" target="_self">Th40</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  <a href="comput_1.html#V5" target="_self">with_the_same_arity</a>  <a href="finseq_2.html#M2">Element</a> of <a href="comput_1.html#K11" target="_self">PrimRec</a>  <a href="finseq_2.html#K3">*</a>  ;<br/>
<a NAME="E9:158"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is    <a href="comput_1.html#M1" target="_self">Element</a> of  <a href="comput_1.html#K11" target="_self">PrimRec</a> 
 
<b>by </b><i><a class="ref" href="comput_1.html#D19" target="_self">Def19</a></i>;<br/>
<a NAME="E10:158"/>
 <a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E1:158"><i><font color="Green">E104</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<b>hence </b><a NAME="E11:158"/>
 <a href="comput_1.html#K16" target="_self">(1,2)-&gt;(1,?,2)</a> <font color="Maroon">c<sub>1</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#E9:158"><i><font color="Green">E106</font></i></a>, <a class="ref" href="comput_1.html#T76" target="_self">Th76</a></i>;<br/>


</div>
