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

<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:3"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:3"/><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>

<a NAME="E3:3"/><i><font color="Green">E24</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/>
<a NAME="E4:3"/><i><font color="Green">E25</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E5:3"/><i><font color="Green">E31</font></i>: 
not for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B</font> iff <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_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"/>
(  ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B</font> &amp; not <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B</font> ) or  ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( not <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B</font> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1"><b>case </b></a><a NAME="E1:3_1_1_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B</font> &amp; not <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:3_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E4:3_1_1_1"/><i><font color="Green">E34</font></i>: 
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B</font>
 ;<br/><b>consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:3_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E7:3_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E8:3_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E9:3_1_1_1"/><i><font color="Green">E39</font></i>: 
not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E10:3_1_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E11:3_1_1_1"/>
<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">z</font>
 <b>by </b><i>, , , <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><b>hence </b><a NAME="E12:3_1_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_2"><b>case </b></a><a NAME="E1:3_1_1_2"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( not <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:3_1_1_2"/><i><font color="Green">E41</font></i>: 
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E4:3_1_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">B</font>
 ;<br/><b>consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:3_1_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E7:3_1_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E8:3_1_1_2"/><i><font color="Green">E45</font></i>: 
<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">z</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E9:3_1_1_2"/><i><font color="Green">E46</font></i>: 
not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E10:3_1_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E11:3_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, , , <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><b>hence </b><a NAME="E12:3_1_1_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:3"/>
contradiction
 ;<br/>


</div>
