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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">X0</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c4">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X0</font>;<br/>







<b>assume </b><a NAME="E1:63"/><i><font color="Green" title="E37">A1</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">C</font> = <font color="Maroon" title="c3">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X0</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E3:63"/><i><font color="Green" title="E38">A2</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">C</font>
 
<b>by </b><i><a class="ref" href="tops_3.html#T56" target="_self" title="TOPS_3:th.56">Th56</a></i>;<br/>
<b>assume </b><a NAME="E4:63"/>
<font color="Maroon" title="c3">A</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>
 ;<br/>

<a NAME="E5:63"/><b>then </b>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">A</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>
 
<b>by </b><i><a class="ref" href="tops_3.html#T35" target="_self" title="TOPS_3:th.35">Th35</a></i>;<br/>
<a NAME="E6:63"/><b>then </b>
(  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">C</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> &amp;  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c4">B</font> )
 
<b>by </b><i><a class="txt" href="tops_3.html#E1:63"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="tops_3.html#E3:63"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="tops_3.html#T59" target="_self" title="TOPS_3:th.59">Th59</a>, <a class="ref" href="tops_1.html#T48" title="TOPS_1:th.48">TOPS_1:48</a></i>;<br/>
<a NAME="E7:63"/><b>then </b>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c4">B</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T79" title="TOPS_1:th.79">TOPS_1:79</a></i>;<br/>
<b>hence </b><a NAME="E8:63"/>
<font color="Maroon" title="c4">B</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T35" target="_self" title="TOPS_3:th.35">Th35</a></i>;<br/>


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