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<b>let </b><font color="Maroon">X</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>thus </b><a NAME="E1:27"/>
( <font color="Maroon">X</font> is <a href="tdlat_3.html#V2" target="_self">anti-discrete</a> implies for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  holds <br/>(  not <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a> or <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Olive">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">X</font> is <a href="tdlat_3.html#V2" target="_self">anti-discrete</a>
 ;<br/>

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E2:27_1"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E3:27_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E4:27_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="tdlat_3.html#T4" target="_self">Th4</a>, </i>;<br/>
<b>hence </b><a NAME="E5:27_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


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

<b>assume </b><a NAME="E2:27"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  holds <br/>(  not <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a> or <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Olive">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 ;<br/>

<a NAME="E3:27"/>
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T5">PRE_TOPC:5</a>, <a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>
<a NAME="E4:27"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">P</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:27_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 ;<br/><b>then reconsider </b><font color="Maroon">C</font> = <font color="Maroon">P</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> ;<br/><a NAME="E3:27_2"/>
<font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><a NAME="E4:27_2"/><b>then </b>
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="tdlat_3.html#T4" target="_self">Th4</a></i>;<br/><b>hence </b><a NAME="E5:27_2"/>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:27"/><b>then </b>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E7:27"/><b>then </b>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E8:27"/>
<font color="Maroon">X</font> is <a href="tdlat_3.html#V2" target="_self">anti-discrete</a>
 <b>by </b><i/>;<br/>


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