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<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> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><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><a NAME="E1:39"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:39_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><a NAME="E2:39_1"/><b>then </b>
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> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:39"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:39_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_2.html#V1">open</a>
 <b>by </b><i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:39"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_2.html#V1">open</a>
 ;<br/>

<a NAME="E4:39"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="topgen_4.html#E1:39"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:39"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="topgen_4.html#E3:39"><i><font color="Green">E20</font></i></a>, <a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>


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