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

<b>set </b><font color="Maroon">S</font> = <font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1_1"/>
( ( <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 ) or ( <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) or <font color="Maroon">D</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_1"><b>suppose </b></a><a NAME="E1:16_1_1_1_1"/>
( <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><b>then </b><font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> = 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">D</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T112">FINSEQ_2:112</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>

;<br/><b>hence </b><a NAME="E3:16_1_1_1_1"/>
<font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> is <a href="amistd_2.html#V1" target="_self">product-like</a>
 <b>by </b><i><a class="ref" href="card_3.html#T19">CARD_3:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_2"><b>suppose </b></a><a NAME="E1:16_1_1_1_2"/>
( <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:16_1_1_1_2"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="catalg_1.html#T7">CATALG_1:7</a></i>;<br/><b>take </b><font color="Maroon">f</font> = 0 <a href="cqc_lang.html#K3">.--&gt;</a> <a href="xboole_0.html#K1">{}</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="amistd_2.html#D2" target="_self">AMISTD_2:def 2</a><br/><a NAME="E3:16_1_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E4:16_1_1_1_2"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:16_1_1_1_2"/>
<font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="card_3.html#T37">CARD_3:37</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_3"><b>suppose </b></a><a NAME="E1:16_1_1_1_3"/>
<font color="Maroon">D</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">D</font> = <font color="Maroon">D</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>set </b><font color="Maroon">S</font> = <font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>;<br/><b>take </b>
 <a href="amistd_2.html#K1" target="_self">PA</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span>
; <i><font color="Red">:: according to </font></i><a class="ref" href="amistd_2.html#D2" target="_self">AMISTD_2:def 2</a><br/><a NAME="E3:16_1_1_1_3"/>
<font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K1" target="_self">PA</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:16_1_1_1_3_1"/>
<font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K1" target="_self">PA</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">x</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="E2:16_1_1_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K1" target="_self">PA</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:16_1_1_1_3_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E5:16_1_1_1_3_1"/><i><font color="Green">E56</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K1" target="_self">PA</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E6:16_1_1_1_3_1"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K1" target="_self">PA</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span></span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K1" target="_self">PA</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E7:16_1_1_1_3_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">D</font> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#D4">FINSEQ_2:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">s</font> being    <a href="pralg_2.html#M1">Element</a> of <font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>;<br/>
<a NAME="E9:16_1_1_1_3_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>
 
;<br/>
<b>then consider </b><font color="Maroon">t</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">D</font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E11:16_1_1_1_3_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>and </b><br/><a NAME="E12:16_1_1_1_3_1"/><i><font color="Green">E62</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/>
<i><font color="Green">E63</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> = 
 <a href="pralg_2.html#K1">DOM</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font>
<b>by </b><i><a class="ref" href="pralg_2.html#D2">PRALG_2:def 2</a></i>
;<br/>
<a NAME="E14:16_1_1_1_3_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E15:16_1_1_1_3_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">g</font> is   <a href="finseq_1.html#NM1">FinSequence</a>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E16:16_1_1_1_3_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_3_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:16_1_1_1_3_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:16_1_1_1_3_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E4:16_1_1_1_3_1_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:16_1_1_1_3_1_2"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K1" target="_self">PA</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:16_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5">pi</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span>,<font color="Maroon">a</font>
 
<b>by </b><i>, , </i>;<br/>
<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:16_1_1_1_3_1_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>
 <b>and </b><br/><a NAME="E9:16_1_1_1_3_1_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon">w</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">D</font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E11:16_1_1_1_3_1_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font>
 <b>and </b><br/><a NAME="E12:16_1_1_1_3_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E13:16_1_1_1_3_1_2"/><i><font color="Green">E82</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">w</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E14:16_1_1_1_3_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">w</font>
 
<b>by </b><i>, , , <a class="ref" href="pralg_2.html#D1">PRALG_2:def 1</a></i>;<br/>
<a NAME="E15:16_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon">w</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">w</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E16:16_1_1_1_3_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font> <b>by </b><i>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E18:16_1_1_1_3_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E19:16_1_1_1_3_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 
<b>by </b><i>, , , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>hence </b><a NAME="E20:16_1_1_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:16_1_1_1_3"/>
<font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K1" target="_self">PA</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
