<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>;<br/>



<b>assume </b><a NAME="E1:5"/><i><font color="Green">E37</font></i>: 
 <a href="arytm_3.html#K13">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>assume </b><a NAME="E2:5"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>assume </b><a NAME="E3:5"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">A</font>, <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Olive">A</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>assume </b><a NAME="E4:5"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">G</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>  st <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> holds <br/> <a href="setfam_1.html#K8">Intersect</a> <font color="Olive">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>set </b><font color="Maroon">O</font> =  <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">F</font>;<br/>
<b>let </b><font color="Maroon">T</font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/>

<b>assume </b><a NAME="E5:5"/><i><font color="Green">E44</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">F</font> )
 ;<br/>

<a NAME="E6:5"/>
<font color="Maroon">T</font> is <a href="pre_topc.html#V2">TopSpace-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<a NAME="E1:5_1"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">F</font>
 
<b>by </b><i>, , <a class="ref" href="setfam_1.html#T49">SETFAM_1:49</a></i>;<br/>
<a NAME="E2:5_1"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">F</font>
 
;<br/>
<b>hence </b><a NAME="E3:5_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a><br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>;<br/><b>assume </b><a NAME="E1:5_1_1"/>
<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 ;<br/><a NAME="E2:5_1_1"/><b>then </b>
 <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#T52">SETFAM_1:52</a></i>;<br/><a NAME="E3:5_1_1"/><b>then </b>
 <a href="setfam_1.html#K8">Intersect</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:5_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">a</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="yellow_8.html#T15">YELLOW_8:15</a></i>;<br/><b>hence </b><a NAME="E5:5_1_1"/>
 <a href="setfam_1.html#K5">union</a> <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">T</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D8">SETFAM_1:def 8</a></i>;<br/></div>
<b>end;</b></div>

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

<b>assume </b><a NAME="E5:5_1"/>
( <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">T</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">T</font> )
 ;<br/>

<a NAME="E6:5_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">b</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D8">SETFAM_1:def 8</a></i>;<br/>
<a NAME="E7:5_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:5_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">b</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T54">XBOOLE_1:54</a></i>;<br/>
<b>hence </b><a NAME="E9:5_1"/>
<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">T</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D8">SETFAM_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:5"/>
<font color="Maroon">T</font> is   <a href="pre_topc.html#NM1">TopSpace</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">T'</font> = <font color="Maroon">T</font> as   <a href="pre_topc.html#NM1">TopSpace</a> ;<br/>
<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>

<a NAME="E9:5"/>
( <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> iff <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>
<a NAME="E10:5"/><b>then </b>
( <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> iff <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">F</font> )
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E11:5"/>
( <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> iff <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 <b>by </b><i>, <a class="ref" href="setfam_1.html#T49">SETFAM_1:49</a></i>;<br/>


</div>
