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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="E1:64"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>assume </b><a NAME="E2:64"/>
<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>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <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> ;<br/>
<a NAME="E4:64"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="tops_3.html#E1:64"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tops_2.html#T33">TOPS_2:33</a></i>;<br/>
<b>assume </b><a NAME="E5:64"/><i><font color="Green">E40</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="E6:64"/><i><font color="Green">E41</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/>

<a NAME="E7:64"/><i><font color="Green">E42</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</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>7</sub></font> =  <a href="tops_1.html#K1">Int</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="txt" href="tops_3.html#E6:64"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:64"/>
 <a href="tops_1.html#K1">Int</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>6</sub></font>
 
<b>by </b><i><a class="txt" href="tops_3.html#E5:64"><i><font color="Green">E40</font></i></a>, <a class="txt" href="tops_3.html#E6:64"><i><font color="Green">E41</font></i></a>, <a class="ref" href="tops_1.html#T48">TOPS_1:48</a></i>;<br/>
<a NAME="E10:64"/><b>then </b><i><font color="Green">E43</font></i>: 
(  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i><a class="txt" href="tops_3.html#E4:64"><i><font color="Green">E39</font></i></a>, <a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/>
<a NAME="E11:64"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="tops_3.html#E1:64"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tsep_1.html#T9">TSEP_1:9</a></i>;<br/>
<a NAME="E12:64"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>7</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#E6:64"><i><font color="Green">E41</font></i></a>, <a class="txt" href="tops_3.html#E7:64"><i><font color="Green">E42</font></i></a>, <a class="ref" href="tops_1.html#T56">TOPS_1:56</a></i>;<br/>
<b>thus </b><a NAME="E13:64"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V1">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="64_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V1">dense</a>
 ;<br/>

<b>assume </b><a NAME="E2:64_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="tops_3.html#V1" target="_self">everywhere_dense</a>
 ;<br/>

<a NAME="E3:64_1"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>5</sub></font> is <a href="tops_1.html#V1">dense</a>
 
<b>by </b><i><a class="ref" href="tops_3.html#T35" target="_self">Th35</a></i>;<br/>
<a NAME="E4:64_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> is <a href="tops_1.html#V1">dense</a>
 
<b>by </b><i><a class="txt" href="tops_3.html#E10:64"><i><font color="Green">E43</font></i></a>, <a class="txt" href="tops_3.html#E1:64_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="tops_3.html#T60" target="_self">Th60</a></i>;<br/>
<a NAME="E5:64_1"/><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#E12:64"><i><font color="Green">E44</font></i></a>, <a class="ref" href="tops_1.html#T79">TOPS_1:79</a></i>;<br/>
<b>hence </b><a NAME="E6:64_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="ref" href="tops_3.html#T35" target="_self">Th35</a></i>;<br/>


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

<b>thus </b><a NAME="E14:64"/>
( <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_1.html#V1">dense</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="tops_3.html#V1" target="_self">everywhere_dense</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="tops_3.html#V1" target="_self">everywhere_dense</a>
 ;<br/>

<a NAME="E2:64_2"/><b>then </b>
<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="txt" href="tops_3.html#E5:64"><i><font color="Green">E40</font></i></a>, <a class="ref" href="tops_3.html#T38" target="_self">Th38</a></i>;<br/>
<b>hence </b><a NAME="E3:64_2"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V1">dense</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T33" target="_self">Th33</a></i>;<br/>

<b>thus </b><a NAME="E4:64_2"/>
<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#E6:64"><i><font color="Green">E41</font></i></a>, <a class="txt" href="tops_3.html#E1:64_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="tops_3.html#T61" target="_self">Th61</a></i>;<br/>


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


</div>
