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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:86"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:86"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="wellord2.html#R2">are_equipotent</a> 
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:86"><i><font color="Green">E56</font></i></a>, <a class="ref" href="card_1.html#T74">CARD_1:74</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:86"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E3:86"><i><font color="Green">E57</font></i></a>, <a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="afinsq_1.html#NM1">XFinSequence</a> <b>by </b><i><a class="txt" href="stirl2_1.html#E5:86"><i><font color="Green">E58</font></i></a>, <a class="ref" href="afinsq_1.html#T7">AFINSQ_1:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="stirl2_1.html#E5:86"><i><font color="Green">E58</font></i></a>, <a class="ref" href="ordinal1.html#D8">ORDINAL1:def 8</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>6</sub></font> =  <a href="stirl2_1.html#K7" target="_self">Sum</a> <font color="Maroon">c<sub>5</sub></font>;<br/>

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

<b>assume </b><a NAME="E8:86"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E9:86"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E35"><i><font color="Green">Lemma24</font></i></a>, <a class="txt" href="stirl2_1.html#E5:86"><i><font color="Green">E58</font></i></a>, <a class="txt" href="stirl2_1.html#E8:86"><i><font color="Green">E59</font></i></a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>
<a NAME="E10:86"/><b>then </b>
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="stirl2_1.html#K8" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> ) )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E5:86"><i><font color="Green">E58</font></i></a>, <a class="txt" href="stirl2_1.html#E8:86"><i><font color="Green">E59</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E11:86"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="stirl2_1.html#T52" target="_self">Th52</a></i>;<br/>


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