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

<b>let </b><font color="Maroon" title="c1">S</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">T</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a><br/> for <font color="Olive" title="b2">K</font> being   <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Olive" title="b1">T</font><br/> for <font color="Olive" title="b3">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Olive" title="b1">T</font> holds <br/> ( <font color="Olive" title="b3">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> iff for <font color="Olive" title="b4">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">T</font>  st <font color="Olive" title="b4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">K</font> holds <br/><font color="Olive" title="b3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )</span><br/><b>let </b><font color="Maroon" title="c2">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">K</font> being   <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c2">T</font><br/> for <font color="Olive" title="b2">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> holds <br/> ( <font color="Olive" title="b2">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> iff for <font color="Olive" title="b3">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>  st <font color="Olive" title="b3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> holds <br/><font color="Olive" title="b2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )</span><br/><b>let </b><font color="Maroon" title="c3">BB</font> be    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c2">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> holds <br/> ( <font color="Olive" title="b1">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> iff for <font color="Olive" title="b2">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>  st <font color="Olive" title="b2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">BB</font> holds <br/><font color="Olive" title="b1">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )</span><br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> iff for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">BB</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )</span><br/>







<a NAME="E1:51"/><i><font color="Green" title="E26">A1</font></i>: 
<font color="Maroon" title="c3">BB</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T</font>
 
<b>by </b><i><a class="ref" href="cantor_1.html#D2" title="CANTOR_1:def.2">CANTOR_1:def 2</a></i>;<br/>
<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">BB</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> ) implies <font color="Maroon" title="c4">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:51_1"/><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c4">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">BB</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a></span><br/><b>let </b><font color="Maroon" title="c5">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">BB</font> implies <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )</span><br/><b>assume </b><a NAME="E2:51_1"/>
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">BB</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a></span><br/><a NAME="E3:51_1"/><b>then </b>
<font color="Maroon" title="c5">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="yellow_9.html#E1:51"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/><a NAME="E4:51_1"/><b>then </b>
<font color="Maroon" title="c5">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T30" title="TOPS_1:th.30">TOPS_1:30</a></i>;<br/><b>hence </b><a NAME="E5:51_1"/>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="yellow_9.html#E1:51_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="pre_topc.html#D12" title="PRE_TOPC:def.12">PRE_TOPC:def 12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:51"/><i><font color="Green" title="E27">A3</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">BB</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a></span><br/>

<b>let </b><font color="Maroon" title="c5">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D12" title="PRE_TOPC:def.12">PRE_TOPC:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> or <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c5">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )</span><br/>

<b>assume </b><a NAME="E4:51"/>
<font color="Maroon" title="c5">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c5">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a></span><br/>

<a NAME="E5:51"/><b>then </b>
<font color="Maroon" title="c5">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></i>;<br/>
<a NAME="E6:51"/><b>then </b><i><font color="Green" title="E28">A4</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">G</font> where <font color="Olive" title="b1">G</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> : ( <font color="Olive" title="b1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">BB</font> &amp; <font color="Olive" title="b1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_8.html#T18" title="YELLOW_8:th.18">YELLOW_8:18</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">F</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">G</font></span>)</span> where <font color="Olive" title="b1">G</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> : ( <font color="Olive" title="b1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">BB</font> &amp; <font color="Olive" title="b1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  ) </span>}</span> ;<br/>
<a NAME="E7:51"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">G</font></span>)</span> where <font color="Olive" title="b1">G</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> : ( <font color="Olive" title="b1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">BB</font> &amp; <font color="Olive" title="b1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">G</font></span>)</span> where <font color="Olive" title="b1">G</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> : ( <font color="Olive" title="b1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">BB</font> &amp; <font color="Olive" title="b1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  ) </span>}</span>  or <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> )</span><br/>

<b>assume </b><a NAME="E1:51_2"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">G</font></span>)</span> where <font color="Olive" title="b1">G</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> : ( <font color="Olive" title="b1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">BB</font> &amp; <font color="Olive" title="b1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><br/>

<a NAME="E2:51_2"/><b>then </b>
 ex <font color="Olive" title="b1">G</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> st <br/>( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">G</font> &amp; <font color="Olive" title="b1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">BB</font> &amp; <font color="Olive" title="b1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 
;<br/>
<b>hence </b><a NAME="E3:51_2"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">F</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">G</font></span>)</span> where <font color="Olive" title="b1">G</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> : ( <font color="Olive" title="b1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">BB</font> &amp; <font color="Olive" title="b1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  ) </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">F</font> = <font color="Maroon" title="c7">F</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<a NAME="E10:51"/>
<font color="Maroon" title="c8">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">F</font> or <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S</font> )</span><br/>

<b>assume </b><a NAME="E1:51_3"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S</font></span><br/>

<b>then consider </b><font color="Maroon" title="c10">G</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font><b> such that </b><br/><a NAME="E3:51_3"/><i><font color="Green" title="E29">A5</font></i>: 
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c10">G</font> &amp; <font color="Maroon" title="c10">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">BB</font> &amp; <font color="Maroon" title="c10">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 ;<br/>
<a NAME="E4:51_3"/>
( <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c10">G</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="yellow_9.html#E3:51"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="yellow_9.html#E3:51_3"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="yellow_9.html#T2" target="_self" title="YELLOW_9:th.2">Th2</a></i>;<br/>
<a NAME="E5:51_3"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c10">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T30" title="TOPS_1:th.30">TOPS_1:30</a></i>;<br/>
<b>hence </b><a NAME="E6:51_3"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S</font>
 <b>by </b><i><a class="txt" href="yellow_9.html#E3:51_3"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E11:51"/><b>then </b><i><font color="Green" title="E29">A6</font></i>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D1" title="PRE_TOPC:def.1">PRE_TOPC:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">BB</font> &amp; $1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  );<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>) <b>-&gt; </b>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> = $1;<br/>
<a NAME="E12:51"/>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">G</font>) where <font color="Olive" title="b1">G</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">G</font>] </span>}</span> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">G</font>)</span>)</span> where <font color="Olive" title="b1">G</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">G</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="yellow_9.html#S4" target="_self" title="YELLOW_9:sch.4">YELLOW_9:sch 4</a>();<br/></i>
<a NAME="E13:51"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="yellow_9.html#E6:51"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="yellow_9.html#E11:51"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E14:51"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T2" target="_self" title="YELLOW_9:th.2">Th2</a></i>;<br/>
<b>hence </b><a NAME="E15:51"/>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c5">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
