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<b>let </b><font color="Maroon" title="c1">X1</font>, <font color="Maroon" title="c2">X2</font>, <font color="Maroon" title="c3">X3</font>, <font color="Maroon" title="c4">X4</font>, <font color="Maroon" title="c5">X5</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>









<b>assume </b><a NAME="E1:3"/><i><font color="Green" title="E2">A1</font></i>: 
( <font color="Maroon" title="c1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X2</font> &amp; <font color="Maroon" title="c2">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X3</font> &amp; <font color="Maroon" title="c3">X3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X4</font> &amp; <font color="Maroon" title="c4">X4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X5</font> &amp; <font color="Maroon" title="c5">X5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X1</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c6">Z</font> = <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span>;<br/>
<a NAME="E2:3"/><i><font color="Green" title="E3">A2</font></i>: 
<font color="Maroon" title="c1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span>
 
<b>by </b><i><a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></i>;<br/>
<a NAME="E3:3"/>
for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:3_1"/><i><font color="Green" title="E4">A3</font></i>: 
<font color="Maroon" title="c7">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_1_1_1"/>
( <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X1</font> or <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X2</font> or <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X3</font> or <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X4</font> or <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">X5</font> )
 </span><b>by </b><i><a class="txt" href="ordinal1.html#E1:3_1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1_1"><b>suppose </b></a><a NAME="E1:3_1_1_1_1"/><i><font color="Green" title="E5">A4</font></i>: 
<font color="Maroon" title="c7">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X1</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c5">X5</font>;<br/><b>thus </b><a NAME="E2:3_1_1_1_1"/>
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> )
 <b>by </b><i><a class="txt" href="ordinal1.html#E1:3"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="ordinal1.html#E1:3_1_1_1_1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1_2"><b>suppose </b></a><a NAME="E1:3_1_1_1_2"/><i><font color="Green" title="E5">A5</font></i>: 
<font color="Maroon" title="c7">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X2</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c1">X1</font>;<br/><b>thus </b><a NAME="E2:3_1_1_1_2"/>
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> )
 <b>by </b><i><a class="txt" href="ordinal1.html#E1:3"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="ordinal1.html#E1:3_1_1_1_2"><i><font color="Green" title="E5">A5</font></i></a>, <a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1_3"><b>suppose </b></a><a NAME="E1:3_1_1_1_3"/><i><font color="Green" title="E5">A6</font></i>: 
<font color="Maroon" title="c7">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X3</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c2">X2</font>;<br/><b>thus </b><a NAME="E2:3_1_1_1_3"/>
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> )
 <b>by </b><i><a class="txt" href="ordinal1.html#E1:3"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="ordinal1.html#E1:3_1_1_1_3"><i><font color="Green" title="E5">A6</font></i></a>, <a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1_4"><b>suppose </b></a><a NAME="E1:3_1_1_1_4"/><i><font color="Green" title="E5">A7</font></i>: 
<font color="Maroon" title="c7">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">X4</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c3">X3</font>;<br/><b>thus </b><a NAME="E2:3_1_1_1_4"/>
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> )
 <b>by </b><i><a class="txt" href="ordinal1.html#E1:3"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="ordinal1.html#E1:3_1_1_1_4"><i><font color="Green" title="E5">A7</font></i></a>, <a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1_5"><b>suppose </b></a><a NAME="E1:3_1_1_1_5"/><i><font color="Green" title="E5">A8</font></i>: 
<font color="Maroon" title="c7">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">X5</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c4">X4</font>;<br/><b>thus </b><a NAME="E2:3_1_1_1_5"/>
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> )
 <b>by </b><i><a class="txt" href="ordinal1.html#E1:3"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="ordinal1.html#E1:3_1_1_1_5"><i><font color="Green" title="E5">A8</font></i></a>, <a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:3_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font>,<font color="Maroon" title="c3">X3</font>,<font color="Maroon" title="c4">X4</font>,<font color="Maroon" title="c5">X5</font></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:3"/>
contradiction
 <b>by </b><i><a class="txt" href="ordinal1.html#E2:3"><i><font color="Green" title="E3">A2</font></i></a>, <a class="ref" href="tarski.html#T7" title="TARSKI:th.7">TARSKI:7</a></i>;<br/>


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