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<b>let </b><font color="Maroon">Y</font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/>

<b>assume </b><a NAME="E1:16_2_1"/>
<font color="Maroon">Y</font> is <a href="tdlat_3.html#V2" target="_self">anti-discrete</a>
 ;<br/>

<a NAME="E2:16_2_1"/><b>then </b><i><font color="Green">E21</font></i>: 
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</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">Y</font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:16_2_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>thus </b><a NAME="E2:16_2_1_1"/>
for <font color="Olive">F</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">Y</font>  st <font color="Olive">F</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> holds <br/> <a href="setfam_1.html#K5">union</a> <font color="Olive">F</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">Y</font>;<br/>

<b>assume </b><a NAME="E1:16_2_1_1_1"/>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:16_2_1_1_1"/><b>then </b>
( <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">F</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">Y</font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T42">ZFMISC_1:42</a></i>;<br/>
<a NAME="E3:16_2_1_1_1"/><b>then </b>
(  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> or  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a>, <a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a>, <a class="ref" href="zfmisc_1.html#T93">ZFMISC_1:93</a></i>;<br/>
<b>hence </b><a NAME="E4:16_2_1_1_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:16_2_1_1"/>
for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> holds <br/><font color="Olive">A</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">B</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:16_2_1_1_2"/>
( <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">Y</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> )
 ;<br/>

<a NAME="E2:16_2_1_1_2"/><b>then </b>
( ( <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">Y</font> ) &amp; ( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> ) )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:16_2_1_1_2"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:16_2_1"/>
<font color="Maroon">Y</font> is <a href="pre_topc.html#V2">TopSpace-like</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>


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