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<b>let </b><font color="Maroon">T</font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">Q</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">M</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>;<br/>







<b>assume </b><a NAME="E1:34"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:34_1"/>
<font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>thus </b><a NAME="E2:34_1"/>
<font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="tops_2.html#K1" target="_self">|</a> <font color="Maroon">M</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_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="E1:34_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">M</font>
 ;<br/>

<a NAME="E2:34_1_1"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</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">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:34_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E5:34_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">Z</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>reconsider </b><font color="Maroon">ZZ</font> = <font color="Maroon">Z</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E7:34_1_1"/>
<font color="Maroon">ZZ</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">M</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="tops_2.html#K1" target="_self">|</a> <font color="Maroon">M</font>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">ZP</font> = <font color="Maroon">ZZ</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">M</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">M</font></span>)</span> ;<br/>
<a NAME="E9:34_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZP</font> &amp; <font color="Maroon">ZP</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="tops_2.html#K1" target="_self">|</a> <font color="Maroon">M</font> )
 
<b>by </b><i>, , , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E10:34_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="tops_2.html#K1" target="_self">|</a> <font color="Maroon">M</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></div><b>end;</b></div>
<b>hence </b><a NAME="E3:34"/>
<font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="tops_2.html#K1" target="_self">|</a> <font color="Maroon">M</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>


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