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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  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <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>let </b><font color="Maroon">c<sub>5</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="E1:66"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:66"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>assume </b><a NAME="E3:66"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>thus </b><a NAME="E4:66"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="tops_3.html#V1" target="_self">everywhere_dense</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="tops_3.html#V1" target="_self">everywhere_dense</a> implies <font color="Maroon">c<sub>4</sub></font> is <a href="tops_3.html#V1" target="_self">everywhere_dense</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:66_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_3.html#V1" target="_self">everywhere_dense</a>
 ;<br/>

<a NAME="E2:66_1"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</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#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E3:66_1"/><b>then </b><i><font color="Green">E42</font></i>: 
(  <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> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i><a class="ref" href="tops_3.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E4:66_1"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="tops_3.html#E1:66"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>assume </b><a NAME="E6:66_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="tops_3.html#V1" target="_self">everywhere_dense</a>
 ;<br/>

<a NAME="E7:66_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="tops_3.html#V1" target="_self">everywhere_dense</a>
 
<b>by </b><i><a class="txt" href="tops_3.html#E2:66_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="tops_3.html#T61" target="_self">Th61</a></i>;<br/>
<a NAME="E8:66_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font> is <a href="tops_3.html#V1" target="_self">everywhere_dense</a>
 
<b>by </b><i><a class="txt" href="tops_3.html#E6:66_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="tops_3.html#T44" target="_self">Th44</a></i>;<br/>
<a NAME="E9:66_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</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="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:66_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="tops_3.html#V1" target="_self">everywhere_dense</a>
 
<b>by </b><i><a class="txt" href="tops_3.html#E3:66_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="tops_3.html#E8:66_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="tops_3.html#E9:66_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="tops_3.html#T62" target="_self">Th62</a></i>;<br/>
<a NAME="E12:66_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="tops_3.html#E3:66"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E13:66_1"/>
<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="txt" href="tops_3.html#E11:66_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="tops_3.html#T38" target="_self">Th38</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:66"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="tops_3.html#V1" target="_self">everywhere_dense</a> implies ( <font color="Maroon">c<sub>3</sub></font> is <a href="tops_3.html#V1" target="_self">everywhere_dense</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="tops_3.html#V1" target="_self">everywhere_dense</a> ) )
 <b>by </b><i><a class="txt" href="tops_3.html#E2:66"><i><font color="Green">E39</font></i></a>, <a class="txt" href="tops_3.html#E3:66"><i><font color="Green">E40</font></i></a>, <a class="ref" href="tops_3.html#T38" target="_self">Th38</a>, <a class="ref" href="tops_3.html#T61" target="_self">Th61</a></i>;<br/>


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