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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="rearran1.html#V1" target="_self">terms've_same_card_as_number</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>assume </b><a NAME="E1:17"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:17"/><b>then </b><i><font color="Green">E17</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="rearran1.html#E4"><i><font color="Green">Lemma4</font></i></a></i>;<br/>
<a NAME="E4:17"/><i><font color="Green">E18</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="rearran1.html#E2:17"><i><font color="Green">E17</font></i></a>, <a class="ref" href="rearran1.html#D1" target="_self">Def1</a></i>;<br/>
<b>assume </b><a NAME="E5:17"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>hence </b><a NAME="E6:17"/>
contradiction
 <b>by </b><i><a class="txt" href="rearran1.html#E1:17"><i><font color="Green">E16</font></i></a>, <a class="txt" href="rearran1.html#E4:17"><i><font color="Green">E18</font></i></a>, <a class="ref" href="card_1.html#T78">CARD_1:78</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>


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