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<b>let </b><font color="Maroon" title="c1">P</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="comput_1.html#V7" title="COMPUT_1:attr.7">primitive-recursively_closed</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="comput_1.html#K3" title="COMPUT_1:func.3">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>;<br/><b>let </b><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">f2</font> be    <a href="comput_1.html#M1" title="COMPUT_1:mode.1">Element</a> of <font color="Maroon" title="c1">P</font>;<br/>



<b>assume </b><a NAME="E1:119"/><i><font color="Green" title="E83">A1</font></i>: 
<span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c3">f2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">i</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="E2:119"/><i><font color="Green" title="E84">A2</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ;<br/>

<a NAME="E3:119"/><i><font color="Green" title="E85">A3</font></i>: 
<font color="Maroon" title="c1">P</font> is <a href="comput_1.html#V6" title="COMPUT_1:attr.6">primitive-recursion_closed</a>
 
<b>by </b><i><a class="ref" href="comput_1.html#D17" target="_self" title="COMPUT_1:def.17">Def17</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:119_1"/>
( <font color="Maroon" title="c2">f1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c2">f1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_1"><b>suppose </b></a><a NAME="E1:119_1_1"/>
<font color="Maroon" title="c2">f1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:119_1_1"/><b>then </b>
 <a href="comput_1.html#K8" title="COMPUT_1:func.8">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c4">i</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="ref" href="comput_1.html#T59" target="_self" title="COMPUT_1:th.59">Th59</a></i>;<br/><b>hence </b><a NAME="E3:119_1_1"/>
 <a href="comput_1.html#K8" title="COMPUT_1:func.8">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="ref" href="comput_1.html#T75" target="_self" title="COMPUT_1:th.75">Th75</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_2"><b>suppose </b></a><a NAME="E1:119_1_2"/>
not <font color="Maroon" title="c2">f1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:119_1_2"/><b>then </b>
 <a href="comput_1.html#K8" title="COMPUT_1:func.8">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c4">i</font> <a href="comput_1.html#R1" title="COMPUT_1:pred.1">is_primitive-recursively_expressed_by</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:119"><i><font color="Green" title="E83">A1</font></i></a>, <a class="txt" href="comput_1.html#E2:119"><i><font color="Green" title="E84">A2</font></i></a>, <a class="ref" href="comput_1.html#T21" target="_self" title="COMPUT_1:th.21">Th21</a>, <a class="ref" href="comput_1.html#T68" target="_self" title="COMPUT_1:th.68">Th68</a></i>;<br/><b>hence </b><a NAME="E3:119_1_2"/>
 <a href="comput_1.html#K8" title="COMPUT_1:func.8">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:119"><i><font color="Green" title="E85">A3</font></i></a>, <a class="ref" href="comput_1.html#D16" target="_self" title="COMPUT_1:def.16">Def16</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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