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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</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>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<b>assume </b><a NAME="E1:10"/><i><font color="Green">E11</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:10"/>
not  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:10"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K2">}</a></span><b> such that </b><br/><a NAME="E6:10"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="card_fin.html#E4:10"><i><font color="Green">E12</font></i></a>, <a class="ref" href="card_fin.html#D1" target="_self">Def1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1"/>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> is <a href="xboole_0.html#V1">empty</a> or  not  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1"><b>suppose </b></a><a NAME="E1:10_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:10_1_1"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="card_1.html#T47">CARD_1:47</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relat_1.html#T65">RELAT_1:65</a>, <a class="ref" href="funct_1.html#T142">FUNCT_1:142</a></i>;<br/><b>hence </b><a NAME="E3:10_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fin.html#E1:10"><i><font color="Green">E11</font></i></a>, <a class="txt" href="card_fin.html#E6:10"><i><font color="Green">E13</font></i></a>, <a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2"><b>suppose </b></a><a NAME="E1:10_1_2"/><i><font color="Green">E14</font></i>: 
not  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:10_1_2"/>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K2">}</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="enumset1.html#T69">ENUMSET1:69</a></i>;<br/><a NAME="E3:10_1_2"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="card_fin.html#E1:10_1_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/><a NAME="E4:10_1_2"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T169">RELAT_1:169</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E5:10_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fin.html#E1:10"><i><font color="Green">E11</font></i></a>, <a class="txt" href="card_fin.html#E6:10"><i><font color="Green">E13</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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