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<b>let </b><font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">Y0</font> be    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y</font>;<br/>



<b>assume </b><a NAME="E1:93"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">Y0</font> is <a href="pre_topc.html#V2">TopSpace-like</a>
 ;<br/>

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

<b>assume </b><a NAME="E2:93"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font>
 ;<br/>

<b>assume </b><a NAME="E3:93"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">A</font> is <a href="tex_4.html#V1" target="_self">anti-discrete</a>
 ;<br/>

<a NAME="E4:93"/>
(  <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">Y0</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y0</font> )
 
<b>by </b><i><a class="ref" href="tex_4.html#T13" target="_self">Th13</a>, <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="E5:93"/><b>then </b><i><font color="Green">E22</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">Y0</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">Y0</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="93_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">D</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:93_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y0</font>
 ;<br/><b>then reconsider </b><font color="Maroon">C</font> = <font color="Maroon">D</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font> ;<br/><b>consider </b><font color="Maroon">E</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E4:93_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">E</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E5:93_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y0</font></span>)</span>
 <b>by </b><i><a class="ref" href="tex_4.html#T17" target="_self">Th17</a>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/><b>reconsider </b><font color="Maroon">E</font> = <font color="Maroon">E</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> ;<br/><a NAME="E7:93_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">E</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tex_4.html#T18" target="_self">Th18</a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:93_1_1_1"/>
( <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">E</font> or <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font> )
 </span><b>by </b><i><a class="ref" href="tex_4.html#T15" target="_self">Th15</a>, <a class="ref" href="tex_4.html#T20" target="_self">Th20</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_1_1_1_1"><b>suppose </b></a><a NAME="E1:93_1_1_1_1"/>
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">E</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:93_1_1_1_1"/>
( <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> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="tex_4.html#T14" target="_self">Th14</a>, <a class="ref" href="tex_4.html#T19" target="_self">Th19</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_1_1_1_2"><b>suppose </b></a><a NAME="E1:93_1_1_1_2"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:93_1_1_1_2"/>
( <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> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="tex_4.html#T14" target="_self">Th14</a>, <a class="ref" href="tex_4.html#T19" target="_self">Th19</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:93_1"/>
<font color="Maroon">D</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">Y0</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tex_4.html#T14" target="_self">Th14</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:93"/><b>then </b>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y0</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">Y0</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="E8:93"/><b>then </b>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y0</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">Y0</font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="tex_4.html#D7" target="_self">Def7</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E9:93"/>
<font color="Maroon">Y0</font> is <a href="tdlat_3.html#V2">anti-discrete</a>
 <b>by </b><i><a class="ref" href="tdlat_3.html#D2">TDLAT_3:def 2</a></i>;<br/>


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