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

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



<b>assume </b><a NAME="E1:89"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">X</font>, <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">Y</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B</font> &amp; <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B</font> holds <br/><font color="Olive">X</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Y</font>
 ;<br/>

<a NAME="E2:89"/><i><font color="Green">E31</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">A</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E3:89"/>
<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">A</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">B</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:89_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">A</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">B</font></span>)</span>
 ;<br/>

<a NAME="E2:89_1"/><b>then </b><i><font color="Green">E32</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<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="E4:89_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E5:89_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:89_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E8:89_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T12" target="_self">Th12</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:89_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y</font>
 ;<br/><a NAME="E2:89_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E21"><i><font color="Green">Lemma42</font></i></a>, <a class="txt" href="zfmisc_1.html#E25"><i><font color="Green">Lemma44</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:89_1_1"/><b>then </b>
( <font color="Maroon">X</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E23"><i><font color="Green">Lemma43</font></i></a>, <a class="txt" href="zfmisc_1.html#E28"><i><font color="Green">Lemma45</font></i></a>, <a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E4:89_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T9" target="_self">Th9</a>, <a class="ref" href="zfmisc_1.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:89_1"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="txt" href="zfmisc_1.html#E23"><i><font color="Green">Lemma43</font></i></a>, <a class="txt" href="zfmisc_1.html#E28"><i><font color="Green">Lemma45</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E11:89_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E25"><i><font color="Green">Lemma44</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:89"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">A</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T10" target="_self">Th10</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
