<?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>assume </b><a NAME="E1:9"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<a NAME="E2:9"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T15">XBOOLE_1:15</a></i>;<br/>
<a NAME="E3:9"/>
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</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="tarski.html#R1">c=</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_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E4:9"/><b>then </b>
(  <a href="setfam_1.html#K1" target="_self">meet</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> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1" target="_self">meet</a> <font color="Maroon">X</font> &amp;  <a href="setfam_1.html#K1" target="_self">meet</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> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1" target="_self">meet</a> <font color="Maroon">Y</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:9"/><b>then </b><i><font color="Green">E19</font></i>: 
 <a href="setfam_1.html#K1" target="_self">meet</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> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" target="_self">meet</a> <font color="Maroon">X</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" target="_self">meet</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E6:9"/>
<span class="p1">(<span class="default"><a href="setfam_1.html#K1" target="_self">meet</a> <font color="Maroon">X</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" target="_self">meet</a> <font color="Maroon">Y</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1" target="_self">meet</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_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="E1:9_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" target="_self">meet</a> <font color="Maroon">X</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" target="_self">meet</a> <font color="Maroon">Y</font></span>)</span>
 ;<br/>

<a NAME="E2:9_1"/><b>then </b><i><font color="Green">E20</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1" target="_self">meet</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1" target="_self">meet</a> <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:9_1_1"/>
<font color="Maroon">Z</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>
 ;<br/><a NAME="E2:9_1_1"/><b>then </b>
( <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or <font color="Maroon">Z</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/><b>hence </b><a NAME="E3:9_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:9_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1" target="_self">meet</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>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:9"/>
 <a href="setfam_1.html#K1" target="_self">meet</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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" target="_self">meet</a> <font color="Maroon">X</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" target="_self">meet</a> <font color="Maroon">Y</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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