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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</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>3</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>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><a NAME="E1:88"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:88"/><i><font color="Green">E54</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>2</sub></font> #)
 ;<br/>

<b>assume </b><a NAME="E3:88"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_3.html#V1" target="_self">everywhere_dense</a>
 ;<br/>

<a NAME="E4:88"/><b>then </b>
(  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V1">dense</a> &amp;  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="tops_3.html#E1:88"><i><font color="Green">E53</font></i></a>, <a class="txt" href="tops_3.html#E2:88"><i><font color="Green">E54</font></i></a>, <a class="ref" href="tops_3.html#T35" target="_self">Th35</a>, <a class="ref" href="tops_3.html#T78" target="_self">Th78</a></i>;<br/>
<a NAME="E5:88"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>4</sub></font> is <a href="tops_1.html#V1">dense</a>
 
<b>by </b><i><a class="txt" href="tops_3.html#E2:88"><i><font color="Green">E54</font></i></a>, <a class="ref" href="tops_3.html#T83" target="_self">Th83</a></i>;<br/>
<b>hence </b><a NAME="E6:88"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="tops_3.html#V1" target="_self">everywhere_dense</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T35" target="_self">Th35</a></i>;<br/>


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