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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:87_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="zfmisc_1.html#K1" target="_self">bool</a> <font color="Maroon">A</font></span>)</span> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:87_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="zfmisc_1.html#K1" target="_self">bool</a> <font color="Maroon">A</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="E3:87_1_1"/><i><font color="Green">E26</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="E4:87_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1" target="_self">bool</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:87_1_1"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T10" target="_self">Th10</a>, </i>;<br/>
<b>hence </b><a NAME="E6:87_1_1"/>
<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="zfmisc_1.html#T9" target="_self">Th9</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:87_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E3:87_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><a NAME="E4:87_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1" target="_self">bool</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:87_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="zfmisc_1.html#K1" target="_self">bool</a> <font color="Maroon">A</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>hence </b><a NAME="E2:87"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" target="_self">bool</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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