<?xml version="1.0"?>
<div class="add">

<a NAME="E1:38_1_1"/>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#T4" title="PENCIL_1:th.4">PENCIL_1:4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:38_1_1"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )
 <span class="kw">and </span><br/><a NAME="E4:38_1_1"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#T2" title="PENCIL_1:th.2">PENCIL_1:2</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E34">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:38_1_1_1"/><span class="lab"><font color="Green" title="E35">A4</font></span>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">t</font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">t</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:38_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_1_1"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#T6" title="ZFMISC_1:th.6">ZFMISC_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E3:38_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_1_1"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E1:38_1_1_1"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="ref" href="zfmisc_1.html#T6" title="ZFMISC_1:th.6">ZFMISC_1:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E6:38_1_1"/>
( <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">t</font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pencil_4.html#K9" title="PENCIL_4:func.9">PairSet</a> (<font color="Maroon" title="c1">t</font>,<font color="Maroon" title="c2">X</font>) &amp; <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">t</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pencil_4.html#K9" title="PENCIL_4:func.9">PairSet</a> (<font color="Maroon" title="c1">t</font>,<font color="Maroon" title="c2">X</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1_1"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="pencil_4.html#D9" target="_self" title="PENCIL_4:def.9">Def9</a></span>;<br/>
<a NAME="E7:38_1_1"/><span class="kw">then </span>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="pencil_4.html#K9" title="PENCIL_4:func.9">PairSet</a> (<font color="Maroon" title="c1">t</font>,<font color="Maroon" title="c2">X</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:38_1_1"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="ref" href="pencil_1.html#T2" title="PENCIL_1:th.2">PENCIL_1:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:38_1_1"/>
 not  <a href="pencil_4.html#K9" title="PENCIL_4:func.9">PairSet</a> (<font color="Maroon" title="c1">t</font>,<font color="Maroon" title="c2">X</font>) is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#T4" title="PENCIL_1:th.4">PENCIL_1:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
