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<b>let </b><font color="Maroon">X</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">X0</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X0</font>;<br/>







<b>assume </b><a NAME="E1:63"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">C</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X0</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E3:63"/><i><font color="Green">E23</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="tops_3.html#D3" target="_self">Def3</a></i>;<br/>
<b>assume </b><a NAME="E4:63"/>
<font color="Maroon">A</font> is <a href="tops_3.html#V1" target="_self">everywhere_dense</a>
 ;<br/>

<a NAME="E5:63"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> is <a href="tops_1.html#V1">dense</a>
 
<b>by </b><i/>;<br/>
<a NAME="E6:63"/><b>then </b>
(  <a href="tops_1.html#K1">Int</a> <font color="Maroon">C</font> is <a href="tops_1.html#V1">dense</a> &amp;  <a href="tops_1.html#K1">Int</a> <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, <a class="ref" href="tops_3.html#T30" target="_self">Th30</a>, <a class="ref" href="tops_3.html#T27" target="_self">Th27</a>, <a class="ref" href="tops_1.html#T48">TOPS_1:48</a></i>;<br/>
<a NAME="E7:63"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">B</font> is <a href="tops_1.html#V1">dense</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T79">TOPS_1:79</a></i>;<br/>
<b>hence </b><a NAME="E8:63"/>
<font color="Maroon">B</font> is <a href="tops_3.html#V1" target="_self">everywhere_dense</a>
 <b>by </b><i/>;<br/>


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