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<div class="add">

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:13_1_1_1"/><i><font color="Green">E5</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="E2:13_1_1_1"/><i><font color="Green">E6</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="E3:13_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/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:13_1_1_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="tex_4.html#E5:13_1_1_1"><i><font color="Green">E7</font></i></a></i>;<br/><a NAME="E7:13_1_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="tex_4.html#E5:13_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E8:13_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>
 <b>by </b><i><a class="txt" href="tex_4.html#E1:13_1_1_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="tex_4.html#E2:13_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="tex_4.html#D1" target="_self">Def1</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:13_1_1"/><i><font color="Green">E5</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#V3">open</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="13_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="pre_topc.html#NM2">Point</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>1</sub></font>;<br/><b>assume </b><a NAME="E1:13_1_1_2"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><b>assume </b><a NAME="E2:13_1_1_2"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>assume </b><a NAME="E3:13_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E4:13_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="tex_4.html#E2:13_1_1_2"><i><font color="Green">E7</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E5:13_1_1_2"/>
<font color="Maroon">c<sub>2</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="tex_4.html#E2:13_1_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="tex_4.html#E1:13_1_1_2"><i><font color="Green">E6</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:13_1_1"/>
<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#D1" target="_self">Def1</a></i>;<br/>


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