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

<b>let </b><font color="Maroon" title="c1">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>let </b><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">f2</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="comput_1.html#NV2" title="COMPUT_1:NV.2">to-naturals</a> <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a> <a href="comput_1.html#V2" title="COMPUT_1:attr.2">from-natural-fseqs</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:149"/><i><font color="Green" title="E98">A1</font></i>: 
<font color="Maroon" title="c2">f1</font> is <a href="comput_1.html#V3" title="COMPUT_1:attr.3">len-total</a>
 <b>and </b><br/><a NAME="E2:149"/><i><font color="Green" title="E99">A2</font></i>: 
<font color="Maroon" title="c3">f2</font> is <a href="comput_1.html#V3" title="COMPUT_1:attr.3">len-total</a>
 <b>and </b><br/><a NAME="E3:149"/><i><font color="Green" title="E100">A3</font></i>: 
 <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c2">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>and </b><br/><a NAME="E4:149"/><i><font color="Green" title="E101">A4</font></i>: 
 <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c3">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">p</font> = <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">i</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> as    <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="c2">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>  <b>by </b><i><a class="txt" href="comput_1.html#E3:149"><i><font color="Green" title="E100">A3</font></i></a></i>;<br/>
<a NAME="E6:149"/><i><font color="Green" title="E102">A5</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> 1,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="funct_7.html#T97" title="FUNCT_7:th.97">FUNCT_7:97</a></i>;<br/>
<a NAME="E7:149"/><i><font color="Green" title="E103">A6</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> 1,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">i</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="funct_7.html#T97" title="FUNCT_7:th.97">FUNCT_7:97</a></i>;<br/>
<a NAME="E8:149"/>
<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#K1" title="NAT_1:func.1">+</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>
 
<b>by </b><i><a class="txt" href="comput_1.html#E3:149"><i><font color="Green" title="E100">A3</font></i></a>, <a class="txt" href="comput_1.html#E4:149"><i><font color="Green" title="E101">A4</font></i></a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="comput_1.html#K9" title="COMPUT_1:func.9">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> = 
<font color="Maroon" title="c3">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> 1,<font color="Maroon" title="c1">i</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comput_1.html#K9" title="COMPUT_1:func.9">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> 1,<font color="Maroon" title="c1">i</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="comput_1.html#E1:149"><i><font color="Green" title="E98">A1</font></i></a>, <a class="txt" href="comput_1.html#E2:149"><i><font color="Green" title="E99">A2</font></i></a>, <a class="txt" href="comput_1.html#E3:149"><i><font color="Green" title="E100">A3</font></i></a>, <a class="txt" href="comput_1.html#E6:149"><i><font color="Green" title="E102">A5</font></i></a>, <a class="ref" href="comput_1.html#T67" target="_self" title="COMPUT_1:th.67">Th67</a></i>
<br/>.= 
<font color="Maroon" title="c3">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comput_1.html#K9" title="COMPUT_1:func.9">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">i</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<b>by </b><i><a class="txt" href="comput_1.html#E7:149"><i><font color="Green" title="E103">A6</font></i></a>, <a class="ref" href="finseq_1.html#D9" title="FINSEQ_1:def.9">FINSEQ_1:def 9</a></i>
;<br/><br/>


</div>
