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<b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">D</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/>



<a NAME="E1:3"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_1"/>
(  not <font color="Maroon">D</font> is <a href="xboole_0.html#V1">empty</a> or <font color="Maroon">D</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1"><b>suppose </b></a><a NAME="E1:3_1_1"/>
not <font color="Maroon">D</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:3_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>
 <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="3_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:3_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">D</font></span>)</span>
 ;<br/><a NAME="E2:3_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><b>hence </b><a NAME="E3:3_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="finseq_2.html#K2">|-&gt;</a> <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:3_1_1"/>
<font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, <a class="ref" href="msscyc_1.html#T1">MSSCYC_1:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_2"><b>suppose </b></a><a NAME="E1:3_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">D</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_1_2_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:3_1_2_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:3_1_2_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="finset_1.html#V1">finite</a>
 <b>by </b><i>, , <a class="ref" href="catalg_1.html#T7">CATALG_1:7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:3_1_2_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:3_1_2_1_1_2"/><b>then </b>
<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"><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/><b>hence </b><a NAME="E3:3_1_2_1_1_2"/>
<font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:3_1_2"/>
<font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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