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<b>let </b><font color="Maroon">P</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="comput_1.html#V8" target="_self">primitive-recursively_closed</a> <a href="subset_1.html#NM2">Subset</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>;<br/><b>let </b><font color="Maroon">f1</font> be    <a href="comput_1.html#M1" target="_self">Element</a> of <font color="Maroon">P</font>, <font color="Maroon">f2</font> be    <a href="comput_1.html#M1" target="_self">Element</a> of <font color="Maroon">P</font>;<br/>



<b>assume </b><a NAME="E1:122"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">f1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">f2</font>
 ;<br/>

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

<b>assume </b><a NAME="E2:122"/><i><font color="Green">E49</font></i>: 
( 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> <span class="p1">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">f1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>

<a NAME="E3:122"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">P</font> is <a href="comput_1.html#V7" target="_self">primitive-recursion_closed</a>
 
<b>by </b><i><a class="ref" href="comput_1.html#T13" target="_self">Th13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:122_1"/>
( <font color="Maroon">f1</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">f1</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_1"><b>suppose </b></a><a NAME="E1:122_1_1"/>
<font color="Maroon">f1</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:122_1_1"/><b>then </b>
 <a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:122_1_1"/>
 <a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="comput_1.html#T15" target="_self">Th15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_2"><b>suppose </b></a><a NAME="E1:122_1_2"/>
not <font color="Maroon">f1</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:122_1_2"/><b>then </b>
 <a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</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#T16" target="_self">Th16</a>, <a class="ref" href="comput_1.html#T17" target="_self">Th17</a>, , </i>;<br/><b>hence </b><a NAME="E3:122_1_2"/>
 <a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="comput_1.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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