<?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">X2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X3</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:49"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">X1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">X2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">X3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">X2</font>,<font color="Maroon">X3</font></span><a href="zfmisc_1.html#K3">:]</a></span>;<br/>

<b>set </b><font color="Maroon">Y'</font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K6" target="_self">`2</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>;<br/>
<b>set </b><font color="Maroon">Y</font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K6" target="_self">`2</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>;<br/>
<b>set </b><font color="Maroon">X'</font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><span class="p3"><a href="tarski.html#K2">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K6" target="_self">`2</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K7" target="_self">`3</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>;<br/>
<b>set </b><font color="Maroon">X</font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><span class="p3"><a href="tarski.html#K2">{</a><span class="default"><span class="p4"><a href="tarski.html#K2">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K6" target="_self">`2</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>,<span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K7" target="_self">`3</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K2">{</a><span class="default"><span class="p4"><a href="tarski.html#K2">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K6" target="_self">`2</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>;<br/>
<i><font color="Green">E40</font></i>: <font color="Maroon">x</font> = 
<span class="p1"><a href="mcart_1.html#K3" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K6" target="_self">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K7" target="_self">`3</a> </span>)</span></span><a href="mcart_1.html#K3" target="_self">]</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K6" target="_self">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K7" target="_self">`3</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>

;<br/>
<b>then </b><font color="Maroon">x</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><span class="p3"><a href="tarski.html#K2">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K6" target="_self">`2</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K7" target="_self">`3</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="tarski.html#D5">TARSKI:def 5</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><span class="p3"><a href="tarski.html#K2">{</a><span class="default"><span class="p4"><a href="tarski.html#K2">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K6" target="_self">`2</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>,<span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K7" target="_self">`3</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K2">{</a><span class="default"><span class="p4"><a href="tarski.html#K2">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K6" target="_self">`2</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="tarski.html#D5">TARSKI:def 5</a></i>
;<br/>
<a NAME="E4:49"/><b>then </b>
( ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a>  or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="mcart_1.html#K6" target="_self">`2</a>  ) implies ( <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><span class="p3"><a href="tarski.html#K2">{</a><span class="default"><span class="p4"><a href="tarski.html#K2">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K6" target="_self">`2</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>,<span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K7" target="_self">`3</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K2">{</a><span class="default"><span class="p4"><a href="tarski.html#K2">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K6" target="_self">`2</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K6" target="_self">`2</a> </span>)</span></span><a href="tarski.html#K2">}</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K6" target="_self">`2</a> </span>)</span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K6" target="_self">`2</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K6" target="_self">`2</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><span class="p3"><a href="tarski.html#K2">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K6" target="_self">`2</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K7" target="_self">`3</a> </span>)</span></span><a href="tarski.html#K2">}</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><span class="p3"><a href="tarski.html#K2">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K6" target="_self">`2</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K7" target="_self">`3</a> </span>)</span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><span class="p3"><a href="tarski.html#K2">{</a><span class="default"><span class="p4"><a href="tarski.html#K2">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K6" target="_self">`2</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>,<span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K7" target="_self">`3</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K2">{</a><span class="default"><span class="p4"><a href="tarski.html#K2">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K6" target="_self">`2</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a> </span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span> ) )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:49"/>
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> <a href="mcart_1.html#K5" target="_self">`1</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> <a href="mcart_1.html#K6" target="_self">`2</a>  )
 <b>by </b><i><a class="ref" href="ordinal1.html#T4">ORDINAL1:4</a></i>;<br/>

<a NAME="E6:49"/>
<font color="Maroon">x</font> <a href="mcart_1.html#K7" target="_self">`3</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="mcart_1.html#K2" target="_self">`2</a> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:49"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> <a href="mcart_1.html#K7" target="_self">`3</a> 
 <b>by </b><i>, </i>;<br/>


</div>
