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

<b>assume </b><a NAME="E1:49"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<b>thus </b><a NAME="E2:49"/>
<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">F</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K2">UNION</a> <font color="Maroon">F</font>,<font color="Maroon">G</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="49_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="E1:49_1"/><i><font color="Green">E39</font></i>: 
<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">F</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">G</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">F</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">G</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1"><b>suppose </b></a><a NAME="E1:49_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">F</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="E3:49_1_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">W</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:49_1_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="topgen_4.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>set </b><font color="Maroon">YW</font> = <font color="Maroon">Y</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">W</font>;<br/><a NAME="E6:49_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E7:49_1_1_1"/>
<font color="Maroon">Y</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">W</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K2">UNION</a> <font color="Maroon">F</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E3"><i><font color="Green">Lemma30</font></i></a>, , <a class="ref" href="setfam_1.html#D4">SETFAM_1:def 4</a></i>;<br/><b>hence </b><a NAME="E8:49_1_1_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"><a href="setfam_1.html#K2">UNION</a> <font color="Maroon">F</font>,<font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="topgen_4.html#E3"><i><font color="Green">Lemma30</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_2"><b>suppose </b></a><a NAME="E1:49_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">G</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="E3:49_1_1_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">W</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:49_1_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="topgen_4.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>set </b><font color="Maroon">YW</font> = <font color="Maroon">W</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font>;<br/><a NAME="E6:49_1_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E7:49_1_1_2"/>
<font color="Maroon">W</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K2">UNION</a> <font color="Maroon">F</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E3"><i><font color="Green">Lemma30</font></i></a>, , <a class="ref" href="setfam_1.html#D4">SETFAM_1:def 4</a></i>;<br/><b>hence </b><a NAME="E8:49_1_1_2"/>
<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"><a href="setfam_1.html#K2">UNION</a> <font color="Maroon">F</font>,<font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="topgen_4.html#E3"><i><font color="Green">Lemma30</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E3:49"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K2">UNION</a> <font color="Maroon">F</font>,<font color="Maroon">G</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">F</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i/>;<br/>


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