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

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</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="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">f1</font> is  <a href="comput_1.html#V2" title="COMPUT_1:attr.2">len-total</a>  &amp; <font color="Olive" title="b2">f2</font> is  <a href="comput_1.html#V2" title="COMPUT_1:attr.2">len-total</a>  &amp;  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Olive" title="b2">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 holds <br/><span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font>,1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&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#K7" title="COMPUT_1:func.7">primrec</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font>,1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c1">i</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span><br/><span class="kw">let </span><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="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</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="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f1</font> is  <a href="comput_1.html#V2" title="COMPUT_1:attr.2">len-total</a>  &amp; <font color="Maroon" title="c3">f2</font> is  <a href="comput_1.html#V2" title="COMPUT_1:attr.2">len-total</a>  &amp;  <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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <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 implies <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&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#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c1">i</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:152"/><span class="lab"><font color="Green" title="E96">A1</font></span>: 
<font color="Maroon" title="c2">f1</font> is  <a href="comput_1.html#V2" title="COMPUT_1:attr.2">len-total</a> 
 <span class="kw">and </span><br/><a NAME="E2:152"/><span class="lab"><font color="Green" title="E97">A2</font></span>: 
<font color="Maroon" title="c3">f2</font> is  <a href="comput_1.html#V2" title="COMPUT_1:attr.2">len-total</a> 
 <span class="kw">and </span><br/><a NAME="E3:152"/><span class="lab"><font color="Green" title="E98">A3</font></span>: 
 <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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E4:152"/><span class="lab"><font color="Green" title="E99">A4</font></span>: 
 <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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&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#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c1">i</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">p</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c1">i</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&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#K2" title="NAT_1:func.2">+</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:152"><span class="lab"><font color="Green" title="E98">A3</font></span></a>, <a class="ref" href="finseq_2.html#T151" title="FINSEQ_2:th.151">FINSEQ_2:151</a></span>;<br/>
<a NAME="E6:152"/><span class="lab"><font color="Green" title="E100">A5</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> 1,<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T97" title="FUNCT_7:th.97">FUNCT_7:97</a></span>;<br/>
<a NAME="E7:152"/><span class="lab"><font color="Green" title="E101">A6</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> 1,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c1">i</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T97" title="FUNCT_7:th.97">FUNCT_7:97</a></span>;<br/>
<a NAME="E8:152"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:152"><span class="lab"><font color="Green" title="E98">A3</font></span></a>, <a class="txt" href="#E4:152"><span class="lab"><font color="Green" title="E99">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> = 
<font color="Maroon" title="c3">f2</font> <a href="recdef_1.html#K1" title="RECDEF_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#K8" title="COMPUT_1:func.8">+*</a> 1,<font color="Maroon" title="c1">i</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> 1,<font color="Maroon" title="c1">i</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:152"><span class="lab"><font color="Green" title="E96">A1</font></span></a>, <a class="txt" href="#E2:152"><span class="lab"><font color="Green" title="E97">A2</font></span></a>, <a class="txt" href="#E3:152"><span class="lab"><font color="Green" title="E98">A3</font></span></a>, <a class="txt" href="#E6:152"><span class="lab"><font color="Green" title="E100">A5</font></span></a>, <a class="ref" href="comput_1.html#T67" target="_self" title="COMPUT_1:th.67">Th67</a></span>
<br/>.= 
<font color="Maroon" title="c3">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&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#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>,1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c1">i</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:152"><span class="lab"><font color="Green" title="E101">A6</font></span></a>, <a class="ref" href="finseq_1.html#D9" title="FINSEQ_1:def.9">FINSEQ_1:def 9</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
