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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:43"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_2.html#V2">closed</a>
 <b>and </b><br/><a NAME="E2:43"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_2.html#V2">closed</a>
 ;<br/>

<a NAME="E3:43"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="topgen_4.html#K3" target="_self">UNION</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:43_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topgen_4.html#K3" target="_self">UNION</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:43_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="setfam_1.html#D4">SETFAM_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="topgen_4.html#E3:43_1"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E5:43_1"/>
( <font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i><a class="txt" href="topgen_4.html#E3:43_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="topgen_4.html#E1:43"><i><font color="Green">E22</font></i></a>, <a class="txt" href="topgen_4.html#E2:43"><i><font color="Green">E23</font></i></a>, <a class="ref" href="tops_2.html#D2">TOPS_2:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:43_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="topgen_4.html#E3:43_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="tops_1.html#T36">TOPS_1:36</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:43"/>
 <a href="topgen_4.html#K3" target="_self">UNION</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> is <a href="tops_2.html#V2">closed</a>
 <b>by </b><i><a class="ref" href="tops_2.html#D2">TOPS_2:def 2</a></i>;<br/>


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