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<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:14_1_1_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="tex_4.html#V1" target="_self">anti-discrete</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>assume </b><a NAME="E2:14_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/><a NAME="E3:14_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</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="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/><a NAME="E4:14_1_1_1"/><b>then </b><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a>  or <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a>  )
 <b>by </b><i><a class="txt" href="tex_4.html#E1:14_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="tex_4.html#D3" target="_self">Def3</a></i>;<br/><b>assume </b><a NAME="E5:14_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E6:14_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="tex_4.html#E4:14_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/><b>hence </b><a NAME="E7:14_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:14_1_1_2"/><i><font color="Green">E6</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>(  not <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> or <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2_1"><b>now </b></a><div class="add"><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>assume </b><a NAME="E1:14_1_1_2_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><a NAME="E2:14_1_1_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T22">PRE_TOPC:22</a></i>;<br/><a NAME="E3:14_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="tex_4.html#E1:14_1_1_2_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/><a NAME="E4:14_1_1_2_1"/><b>then </b><i><font color="Green">E8</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a>  or <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a>  )
 <b>by </b><i><a class="txt" href="tex_4.html#E1:14_1_1_2"><i><font color="Green">E6</font></i></a></i>;<br/><b>assume </b><a NAME="E5:14_1_1_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E6:14_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="tex_4.html#E4:14_1_1_2_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/><b>hence </b><a NAME="E7:14_1_1_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:14_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tex_4.html#V1" target="_self">anti-discrete</a>
 <b>by </b><i><a class="ref" href="tex_4.html#D3" target="_self">Def3</a></i>;<br/></div>
<b>end;</b></div>


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