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<b>let </b><font color="Maroon" title="c4">g1</font>, <font color="Maroon" title="c5">g2</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> ;<br/>

<b>given </b><font color="Maroon" title="c6">G1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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> , <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> such that </b><a NAME="E2:97_1_2"/><i><font color="Green" title="E59">A7</font></i>: 
<font color="Maroon" title="c4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c6">G1</font>
 <b>and </b><br/><a NAME="E3:97_1_2"/><i><font color="Green" title="E60">A8</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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>  holds  <font color="Maroon" title="c6">G1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comput_1.html#K8" title="COMPUT_1:func.8">primrec</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">i</font>,<font color="Olive" title="b1">p</font>
 ;<br/>

<b>given </b><font color="Maroon" title="c7">G2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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> , <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> such that </b><a NAME="E5:97_1_2"/><i><font color="Green" title="E61">A9</font></i>: 
<font color="Maroon" title="c5">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c7">G2</font>
 <b>and </b><br/><a NAME="E6:97_1_2"/><i><font color="Green" title="E62">A10</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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>  holds  <font color="Maroon" title="c7">G2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comput_1.html#K8" title="COMPUT_1:func.8">primrec</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">i</font>,<font color="Olive" title="b1">p</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">p</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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> ;<br/><b>thus </b><font color="Maroon" title="c6">G1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font> = 
 <a href="comput_1.html#K8" title="COMPUT_1:func.8">primrec</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c8">p</font>
<b>by </b><i><a class="txt" href="comput_1.html#E3:97_1_2"><i><font color="Green" title="E60">A8</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c7">G2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font>
<b>by </b><i><a class="txt" href="comput_1.html#E6:97_1_2"><i><font color="Green" title="E62">A10</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:97_1_2"/>
<font color="Maroon" title="c4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E2:97_1_2"><i><font color="Green" title="E59">A7</font></i></a>, <a class="txt" href="comput_1.html#E5:97_1_2"><i><font color="Green" title="E61">A9</font></i></a>, <a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


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