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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  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y</font>;<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="E1:66"/><i><font color="Green">E17</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/>

<a NAME="E2:66"/>
(  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y0</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font> &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> )
 
;<br/>
<a NAME="E3:66"/><b>then </b><i><font color="Green">E22</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:66_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">A</font> is <a href="tex_2.html#V3" target="_self">discrete</a>
 ;<br/><a NAME="E2:66_1"/>
 <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y0</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">C</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:66_1_1_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font>
 ;<br/><b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">C</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font> ;<br/><b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">B</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>by </b><i><a class="txt" href="tex_2.html#E3:66"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>consider </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E5:66_1_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E6:66_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 <b>by </b><i>, , <a class="txt" href="tex_2.html#E1:66"><i><font color="Green">E17</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">G</font> = <font color="Maroon">G</font>;<br/><b>thus </b><a NAME="E1:66_1_1_1_1"/>
<font color="Maroon">G</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="txt" href="tex_2.html#E1:66_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><b>thus </b><a NAME="E2:66_1_1_1_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</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>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:66_1_1_1"/>
<font color="Maroon">C</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="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:66_1_1"/>
 <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font> <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="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:66_1"/><b>then </b>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y0</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E4:66_1"/>
<font color="Maroon">Y0</font> is <a href="tdlat_3.html#V1">discrete</a>
 <b>by </b><i><a class="ref" href="tdlat_3.html#D1">TDLAT_3:def 1</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:66_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">Y0</font> is <a href="tdlat_3.html#V1">discrete</a>
 ;<br/><a NAME="E2:66_2"/><b>then </b>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y0</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font>
 <b>by </b><i><a class="ref" href="tdlat_3.html#D1">TDLAT_3:def 1</a></i>;<br/><a NAME="E3:66_2"/><b>then </b>
( 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> &amp; ( for <font color="Olive">F</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">Y0</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">Y0</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">Y0</font> ) &amp; ( for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</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">Y0</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">Y0</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">Y0</font> ) )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D1">ZFMISC_1:def 1</a></i>;<br/><a NAME="E4:66_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">Y0</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">D</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/><b>assume </b><a NAME="E1:66_2_1"/>
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 ;<br/><b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">D</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font> <b>by </b><i/>;<br/><a NAME="E3:66_2_1"/>
<font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, , <a class="ref" href="tdlat_3.html#T17">TDLAT_3:17</a></i>;<br/><a NAME="E4:66_2_1"/><b>then </b>
<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">Y0</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E6:66_2_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">G</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="E7:66_2_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</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="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/><b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> ;<br/><b>take </b><font color="Maroon">G</font> = <font color="Maroon">G</font>;<br/><b>thus </b><a NAME="E9:66_2_1"/>
<font color="Maroon">G</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/><b>thus </b><a NAME="E10:66_2_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:66_2"/>
<font color="Maroon">A</font> is <a href="tex_2.html#V3" target="_self">discrete</a>
 <b>by </b><i><a class="txt" href="tex_2.html#E1:66"><i><font color="Green">E17</font></i></a></i>;<br/></div>
<b>end;</b></div>


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