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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="seqm_3.html#V6">to-naturals</a> <a href="comput_1.html#V2" target="_self">from-natural-fseqs</a> <a href="comput_1.html#V4" target="_self">homogeneous</a> <a href="funct_1.html#NM1">Function</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:110"/><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E2:110"/><i><font color="Green">E75</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>
<b>take </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1; <i><font color="Red">:: according to </font></i><a class="ref" href="comput_1.html#D12" target="_self">COMPUT_1:def 12</a><br/>

<b>thus </b><a NAME="E3:110"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="comput_1.html#T60" target="_self">Th60</a></i>;<br/>

<b>thus </b><a NAME="E4:110"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="comput_1.html#E2:110"><i><font color="Green">E75</font></i></a></i>;<br/>

<b>thus </b><a NAME="E5:110"/>
<span class="p1">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>thus </b><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>

<br/>.= 
 <a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="comput_1.html#E1:110"><i><font color="Green">E74</font></i></a></i>
;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E7:110"/><i><font color="Green">E76</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E8:110"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>5</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E9:110"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E2:110"><i><font color="Green">E75</font></i></a>, <a class="txt" href="comput_1.html#E7:110"><i><font color="Green">E76</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E10:110"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>1</sub></font>,0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> iff  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="comput_1.html#E8:110"><i><font color="Green">E77</font></i></a>, <a class="txt" href="comput_1.html#E81"><i><font color="Green">Lemma70</font></i></a></i>;<br/>

<b>thus </b><a NAME="E11:110"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>1</sub></font>,0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> implies <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="comput_1.html#E8:110"><i><font color="Green">E77</font></i></a>, <a class="txt" href="comput_1.html#E9:110"><i><font color="Green">E78</font></i></a>, <a class="txt" href="comput_1.html#E81"><i><font color="Green">Lemma70</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>thus </b><a NAME="E12:110"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> iff ( <font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> ) )
 <b>by </b><i><a class="txt" href="comput_1.html#E8:110"><i><font color="Green">E77</font></i></a>, <a class="txt" href="comput_1.html#E9:110"><i><font color="Green">E78</font></i></a>, <a class="txt" href="comput_1.html#E81"><i><font color="Green">Lemma70</font></i></a></i>;<br/>

<b>thus </b><a NAME="E13:110"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> implies <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="comput_1.html#E8:110"><i><font color="Green">E77</font></i></a>, <a class="txt" href="comput_1.html#E9:110"><i><font color="Green">E78</font></i></a>, <a class="txt" href="comput_1.html#E81"><i><font color="Green">Lemma70</font></i></a></i>;<br/>


</div>
