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<b>set </b><font color="Maroon" title="c1">X</font> =  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>
<b>thus </b><a NAME="E1:112"/>
( 0 <a href="comput_1.html#K5" title="COMPUT_1:func.5">const</a> 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; 1 <a href="comput_1.html#K6" title="COMPUT_1:func.6">succ</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; ( for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">i</font> being   <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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Olive" title="b1">n</font> <a href="comput_1.html#K7" title="COMPUT_1:func.7">proj</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ) )
 <b>by </b><i><a class="ref" href="comput_1.html#T34" target="_self" title="COMPUT_1:th.34">Th34</a>, <a class="ref" href="comput_1.html#T37" target="_self" title="COMPUT_1:th.37">Th37</a>, <a class="ref" href="comput_1.html#T39" target="_self" title="COMPUT_1:th.39">Th39</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="comput_1.html#D17" target="_self" title="COMPUT_1:def.17">COMPUT_1:def 17</a><br/>

<b>thus </b><a NAME="E2:112"/>
for <font color="Olive" title="b1">f</font> being   <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">F</font> being <a href="comput_1.html#V4" title="COMPUT_1:attr.4">with_the_same_arity</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Olive" title="b1">f</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="Olive" title="b2">F</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Olive" title="b2">F</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="comput_1.html#T46" target="_self" title="COMPUT_1:th.46">Th46</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="comput_1.html#D15" target="_self" title="COMPUT_1:def.15">COMPUT_1:def 15</a><br/>

<b>let </b><font color="Maroon" title="c2">g</font> be    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="comput_1.html#D16" target="_self" title="COMPUT_1:def.16">COMPUT_1:def 16</a><br/>

<b>thus </b><a NAME="E3:112"/>
for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being   <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b3">i</font> being   <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>   st <font color="Maroon" title="c2">g</font> <a href="comput_1.html#R1" title="COMPUT_1:pred.1">is_primitive-recursively_expressed_by</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font>,<font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b1">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Olive" title="b2">f2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon" title="c2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/>


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