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

<i><font color="Green">E42</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K2">|-&gt;</a> <a href="numbers.html#K5">NAT</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> = 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K2">|-&gt;</a> <a href="numbers.html#K5">NAT</a> </span>)</span>
<b>by </b><i><a class="ref" href="pralg_3.html#D2">PRALG_3:def 2</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</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="funct_6.html#T9">FUNCT_6:9</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_2"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="pralg_3.html#K3">proj</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K2">|-&gt;</a> <a href="numbers.html#K5">NAT</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:71_1_1_2"/><i><font color="Green">E43</font></i>: 
 <a href="pralg_3.html#K3">proj</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K2">|-&gt;</a> <a href="numbers.html#K5">NAT</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font> is <a href="comput_1.html#V4" target="_self">homogeneous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="finseq_1.html#NM1">FinSequence</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="comput_1.html#D4" target="_self">COMPUT_1:def 4</a><br/>

<b>assume </b><a NAME="E1:71_1_1_2_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K2">|-&gt;</a> <a href="numbers.html#K5">NAT</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K2">|-&gt;</a> <a href="numbers.html#K5">NAT</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 ;<br/>

<b>thus </b> <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="comput_1.html#E1:71_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="comput_1.html#E1:71_1_1_2_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="comput_1.html#E1:71_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="comput_1.html#E1:71_1_1_2_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
;<br/>


</div><b>end;</b></div><a NAME="E2:71_1_1_2"/><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K2">|-&gt;</a> <a href="numbers.html#K5">NAT</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E1:71_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="msualg_1.html#T12">MSUALG_1:12</a></i>;<br/><a NAME="E3:71_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K2">|-&gt;</a> <a href="numbers.html#K5">NAT</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:71_1_1_2_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K2">|-&gt;</a> <a href="numbers.html#K5">NAT</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:71_1_1_2_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K2">|-&gt;</a> <a href="numbers.html#K5">NAT</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K2">|-&gt;</a> <a href="numbers.html#K5">NAT</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="comput_1.html#E1:71_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="comput_1.html#E3:71_1_1_2_2"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E6:71_1_1_2_2"/><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K2">|-&gt;</a> <a href="numbers.html#K5">NAT</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E3:71_1_1_2_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="pralg_3.html#D2">PRALG_3:def 2</a></i>;<br/>
<a NAME="E7:71_1_1_2_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> or  not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> )
 
;<br/>
<a NAME="E8:71_1_1_2_2"/><b>then </b>
( ( <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  ) or <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E9:71_1_1_2_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E3:71_1_1_2_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="comput_1.html#E6:71_1_1_2_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="card_1.html#T51">CARD_1:51</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="pralg_3.html#K3">proj</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K2">|-&gt;</a> <a href="numbers.html#K5">NAT</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font> as   <a href="partfun1.html#NM1">PartFunc</a> of <a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a> , <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="comput_1.html#E2:71_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/><a NAME="E5:71_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> is    <a href="rfunct_3.html#M2">Element</a> of  <a href="rfunct_3.html#K4">PFuncs</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="finseq_2.html#K3">*</a> </span>)</span>,<a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="partfun1.html#T119">PARTFUN1:119</a></i>;<br/><a NAME="E6:71_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E1:71_1_1_2"><i><font color="Green">E43</font></i></a></i>;<br/><b>hence </b><a NAME="E7:71_1_1_2"/>
 <a href="pralg_3.html#K3">proj</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K2">|-&gt;</a> <a href="numbers.html#K5">NAT</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font> is    <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> 
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:71_1_1"/>
 <a href="pralg_3.html#K3">proj</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K2">|-&gt;</a> <a href="numbers.html#K5">NAT</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font> is  <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/>


</div>
