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

<b>let </b><font color="Maroon">X1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Y1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Y2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/>















<b>assume </b><a NAME="E1:17"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">X1</font>,<font color="Maroon">Y1</font> <a href="wellord2.html#R2">are_equipotent</a>  &amp; <font color="Maroon">X2</font>,<font color="Maroon">Y2</font> <a href="wellord2.html#R2">are_equipotent</a>  &amp; <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y2</font> )
 ;<br/>

<a NAME="E2:17"/>
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y1</font></span><a href="tarski.html#K1">}</a></span> <a href="wellord2.html#R2">are_equipotent</a>  &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x2</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y2</font></span><a href="tarski.html#K1">}</a></span> <a href="wellord2.html#R2">are_equipotent</a>  )
 
<b>by </b><i><a class="ref" href="card_1.html#T48">CARD_1:48</a></i>;<br/>
<a NAME="E3:17"/><b>then </b><i><font color="Green">E36</font></i>: 
( <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X1</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">Y1</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y1</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="wellord2.html#R2">are_equipotent</a>  &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X2</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x2</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">Y2</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y2</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="wellord2.html#R2">are_equipotent</a>  )
 
<b>by </b><i>, </i>;<br/>
<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_1"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X1</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X2</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x2</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:17_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X1</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X2</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x2</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:17_1"/>
( <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x2</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/><a NAME="E5:17_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:17_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_2"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">Y1</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y1</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">Y2</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y2</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:17_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">Y1</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y1</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">Y2</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y2</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:17_2"/>
( <font color="Maroon">y</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y1</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">y</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y2</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/><a NAME="E5:17_2"/><b>then </b>
( <font color="Maroon">y</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">y</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:17_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:17"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X1</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X2</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x2</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">Y1</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y1</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">Y2</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y2</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="wellord2.html#R2">are_equipotent</a> 
 <b>by </b><i>, , <a class="ref" href="card_1.html#T58">CARD_1:58</a></i>;<br/>

<b>hence </b><a NAME="E7:17"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X1</font>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X2</font>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x2</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">Y1</font>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y1</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">Y2</font>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y2</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>


</div>
