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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>thus </b><a NAME="E1:25"/>
 <a href="card_3.html#K5" target="_self">pi</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="card_3.html#K5" target="_self">pi</a> <font color="Maroon">X</font>,<font color="Maroon">x</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="card_3.html#K5" target="_self">pi</a> <font color="Maroon">Y</font>,<font color="Maroon">x</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:25_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5" target="_self">pi</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">x</font>
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:25_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:25_1"/>
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:25_1"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5" target="_self">pi</a> <font color="Maroon">X</font>,<font color="Maroon">x</font> or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5" target="_self">pi</a> <font color="Maroon">Y</font>,<font color="Maroon">x</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:25_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="card_3.html#K5" target="_self">pi</a> <font color="Maroon">X</font>,<font color="Maroon">x</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="card_3.html#K5" target="_self">pi</a> <font color="Maroon">Y</font>,<font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:25"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="card_3.html#K5" target="_self">pi</a> <font color="Maroon">X</font>,<font color="Maroon">x</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="card_3.html#K5" target="_self">pi</a> <font color="Maroon">Y</font>,<font color="Maroon">x</font></span>)</span>
 ;<br/>

<a NAME="E3:25"/><b>then </b><i><font color="Green">E47</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5" target="_self">pi</a> <font color="Maroon">X</font>,<font color="Maroon">x</font> or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5" target="_self">pi</a> <font color="Maroon">Y</font>,<font color="Maroon">x</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><i><font color="Green">E54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5" target="_self">pi</a> <font color="Maroon">X</font>,<font color="Maroon">x</font>
 ;<br/><b>then consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:25_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i/>;<br/><a NAME="E4:25_2"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E5:25_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5" target="_self">pi</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_3"/>
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5" target="_self">pi</a> <font color="Maroon">X</font>,<font color="Maroon">x</font>
 ;<br/><b>then consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:25_3"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:25_3"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E5:25_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5" target="_self">pi</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:25"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5" target="_self">pi</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div>
