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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:1"/><i><font color="Green" title="E2">A1</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; (  not  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c1">n</font> or  not <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">b</font></span>)</span> ) )
 ;<br/>

<a NAME="E2:1"/><b>then </b>
( <font color="Maroon" title="c1">n</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">b</font></span>)</span> or  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T26" title="ORDINAL1:th.26">ORDINAL1:26</a></i>;<br/>
<a NAME="E3:1"/><b>then </b>
( <font color="Maroon" title="c1">n</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">b</font></span>)</span> or  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></i>;<br/>
<a NAME="E4:1"/><b>then </b><i><font color="Green" title="E3">A2</font></i>: 
( <font color="Maroon" title="c1">n</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">b</font></span>)</span> or  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a></i>;<br/>
<a NAME="E5:1"/><i><font color="Green" title="E4">A3</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c3">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c2">a</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c2">a</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">b</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E6:1"/><b>then </b>
( <font color="Maroon" title="c1">n</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">b</font></span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="txt" href="combgras.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/>
<a NAME="E7:1"/><b>then </b><i><font color="Green" title="E5">A4</font></i>: 
( (  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">b</font></span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">b</font> ) or (  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">a</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">b</font> ) )
 
<b>by </b><i><a class="txt" href="combgras.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="combgras.html#E4:1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E8:1"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a></i>;<br/>
<a NAME="E9:1"/><b>then </b><i><font color="Green" title="E6">A5</font></i>: 
( <font color="Maroon" title="c2">a</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  &amp; <font color="Maroon" title="c3">b</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  )
 
<b>by </b><i><a class="txt" href="combgras.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="card_4.html#T4" title="CARD_4:th.4">CARD_4:4</a></i>;<br/>
<a NAME="E10:1"/><b>then </b>
( <font color="Maroon" title="c2">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">b</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">b</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  )
 
<b>by </b><i><a class="txt" href="combgras.html#E5:1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a>, <a class="ref" href="finset_1.html#T14" title="FINSET_1:th.14">FINSET_1:14</a></i>;<br/>
<a NAME="E11:1"/><b>then </b>
( ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">b</font> ) or ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">b</font> ) )
 
<b>by </b><i><a class="txt" href="combgras.html#E5:1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="combgras.html#E7:1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="txt" href="combgras.html#E9:1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a></i>;<br/>
<b>hence </b><a NAME="E12:1"/>
contradiction
 <b>by </b><i><a class="txt" href="combgras.html#E1:1"><i><font color="Green" title="E2">A1</font></i></a></i>;<br/>


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