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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</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="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:24"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="hallmar1.html#V1" target="_self">Hall</a>
 ;<br/>

<b>assume </b><a NAME="E2:24"/>
not <font color="Maroon">c<sub>2</sub></font> is <a href="relat_1.html#V2">non-empty</a>
 ;<br/>

<a NAME="E3:24"/><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">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#D9">RELAT_1:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:24"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span>;<br/>
<a NAME="E6:24"/><i><font color="Green">E21</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="hallmar1.html#E5:24"><i><font color="Green">E20</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E7:24"/><i><font color="Green">E22</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="hallmar1.html#K1" target="_self">union</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="hallmar1.html#E5:24"><i><font color="Green">E20</font></i></a>, <a class="ref" href="hallmar1.html#T5" target="_self">Th5</a>, <a class="ref" href="card_1.html#T78">CARD_1:78</a></i>;<br/>
<a NAME="E8:24"/>
 <a href="card_1.html#K4">card</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="card_2.html#T60">CARD_2:60</a></i>;<br/>
<b>hence </b><a NAME="E9:24"/>
contradiction
 <b>by </b><i><a class="txt" href="hallmar1.html#E1:24"><i><font color="Green">E19</font></i></a>, <a class="txt" href="hallmar1.html#E6:24"><i><font color="Green">E21</font></i></a>, <a class="txt" href="hallmar1.html#E7:24"><i><font color="Green">E22</font></i></a>, <a class="ref" href="hallmar1.html#D4" target="_self">Def4</a></i>;<br/>


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