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<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">R</font> be   <a href="relat_1.html#NM1">Relation</a>;<br/>





<b>thus </b><a NAME="E1:17"/>
<font color="Maroon">R</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="relat_1.html#K9">.:</a> <font color="Olive">f</font></span>)</span> where <font color="Olive">f</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font> : <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</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="17_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="E1:17_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span>
 ;<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="E3:17_1"/><i><font color="Green">E21</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#D13">RELAT_1:def 13</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="E5:17_1"/><i><font color="Green">E22</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>set </b><font color="Maroon">Z</font> = <font color="Maroon">R</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">Y</font>;<br/>
<a NAME="E6:17_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">R</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="relat_1.html#K9">.:</a> <font color="Olive">f</font></span>)</span> where <font color="Olive">f</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font> : <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  )
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#D13">RELAT_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E7:17_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="relat_1.html#K9">.:</a> <font color="Olive">f</font></span>)</span> where <font color="Olive">f</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font> : <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</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="E2:17"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="relat_1.html#K9">.:</a> <font color="Olive">f</font></span>)</span> where <font color="Olive">f</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font> : <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:17"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">y</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> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="relat_1.html#K9">.:</a> <font color="Olive">f</font></span>)</span> where <font color="Olive">f</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font> : <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E6:17"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:17"/><i><font color="Green">E26</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> )
 <b>by </b><i>, , <a class="ref" href="relat_1.html#D13">RELAT_1:def 13</a></i>;<br/>
<a NAME="E9:17"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a>, , </i>;<br/>
<b>hence </b><a NAME="E10:17"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="relat_1.html#D13">RELAT_1:def 13</a></i>;<br/>


</div>
