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<b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:82"/><i><font color="Green">E55</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="finseq_2.html#M1">FinSequenceSet</a> of <font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">a<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means verum;<br/>
<a NAME="E2:82"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  is <a href="card_4.html#V1" target="_self">countable</a>
 
<b>from </b><i><a class="ref" href="card_4.html#S1" target="_self">CARD_4:sch 1</a>();<br/></i>
<a NAME="E3:82"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span>  <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K3">alef</a> 0
 
<b>by </b><i><a class="ref" href="card_4.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E4:82"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span>  holds <br/> <a href="card_1.html#K1">Card</a> <font color="Olive">b<sub>1</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:82_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:82_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>
<a NAME="E4:82_1"/>
0 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></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/>
<a NAME="E5:82_1"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default">0 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="finset_1.html#V1">finite</a> &amp; not  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i><a class="txt" href="card_4.html#E1:82"><i><font color="Green">E55</font></i></a>, <a class="ref" href="card_4.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E6:82_1"/><b>then </b>
( (  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default">0 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 ) or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="card_4.html#T13" target="_self">Th13</a></i>;<br/>
<b>hence </b><a NAME="E7:82_1"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>3</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="card_4.html#E1:82"><i><font color="Green">E55</font></i></a>, <a class="txt" href="card_4.html#E3:82_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="card_4.html#T86" target="_self">Th86</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:82"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> </span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="card_2.html#K2">*`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="card_4.html#E3:82"><i><font color="Green">E56</font></i></a>, <a class="ref" href="card_4.html#T60" target="_self">Th60</a></i>;<br/>
<a NAME="E6:82"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_2.html#K3">*</a>  <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T128">FINSEQ_2:128</a></i>;<br/>
<a NAME="E7:82"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_2.html#K3">*</a> </span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="card_4.html#E1:82"><i><font color="Green">E55</font></i></a>, <a class="txt" href="card_4.html#E5:82"><i><font color="Green">E57</font></i></a>, <a class="ref" href="card_4.html#T84" target="_self">Th84</a></i>;<br/>
<a NAME="E8:82"/>
1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span> 
 
;<br/>
<a NAME="E9:82"/><b>then </b>
1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i><a class="txt" href="card_4.html#E6:82"><i><font color="Green">E58</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E10:82"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default">1 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_2.html#K3">*</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<a NAME="E11:82"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_2.html#K3">*</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="card_4.html#T56" target="_self">Th56</a></i>;<br/>
<b>hence </b><a NAME="E12:82"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_2.html#K3">*</a> </span>)</span>
 <b>by </b><i><a class="txt" href="card_4.html#E7:82"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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