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

<b>set </b><font color="Maroon" title="c2">T</font> =  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="complsp1.html#K22" title="COMPLSP1:func.22">ComplexOpenSets</a> <font color="Maroon" title="c1">n</font></span>)</span> #);<br/>
<a NAME="E1:109_1_1"/>
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="complsp1.html#K22" title="COMPLSP1:func.22">ComplexOpenSets</a> <font color="Maroon" title="c1">n</font></span>)</span> #) is <a href="pre_topc.html#V2" title="PRE_TOPC:attr.2">TopSpace-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c3">z</font> =  <a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="ref" href="zfmisc_1.html#D1" title="ZFMISC_1:def.1">ZFMISC_1:def 1</a></i>;<br/>
<a NAME="E2:109_1_1_1"/>
<font color="Maroon" title="c3">z</font> is <a href="complsp1.html#V1" title="COMPLSP1:attr.1">open</a>
 
<b>by </b><i><a class="ref" href="complsp1.html#T77" target="_self" title="COMPLSP1:th.77">Th77</a></i>;<br/>
<b>hence </b><a NAME="E3:109_1_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="complsp1.html#K22" title="COMPLSP1:func.22">ComplexOpenSets</a> <font color="Maroon" title="c1">n</font></span>)</span> #) <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 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="complsp1.html#K22" title="COMPLSP1:func.22">ComplexOpenSets</a> <font color="Maroon" title="c1">n</font></span>)</span> #)
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D1" title="PRE_TOPC:def.1">PRE_TOPC:def 1</a><br/>

<b>thus </b><a NAME="E4:109_1_1_1"/>
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="complsp1.html#K22" title="COMPLSP1:func.22">ComplexOpenSets</a> <font color="Maroon" title="c1">n</font></span>)</span> #)  st <font color="Olive" title="b1">A</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 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="complsp1.html#K22" title="COMPLSP1:func.22">ComplexOpenSets</a> <font color="Maroon" title="c1">n</font></span>)</span> #) holds <br/> <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">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 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="complsp1.html#K22" title="COMPLSP1:func.22">ComplexOpenSets</a> <font color="Maroon" title="c1">n</font></span>)</span> #)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">A</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="complsp1.html#K22" title="COMPLSP1:func.22">ComplexOpenSets</a> <font color="Maroon" title="c1">n</font></span>)</span> #);<br/>

<b>assume </b><a NAME="E1:109_1_1_1_1"/><i><font color="Green" title="E85">A1</font></i>: 
<font color="Maroon" title="c4">A</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 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="complsp1.html#K22" title="COMPLSP1:func.22">ComplexOpenSets</a> <font color="Maroon" title="c1">n</font></span>)</span> #)
 ;<br/>

<a NAME="E2:109_1_1_1_1"/><i><font color="Green" title="E86">A2</font></i>: 
for <font color="Olive" title="b1">B</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> holds <br/><font color="Olive" title="b1">B</font> is <a href="complsp1.html#V1" title="COMPLSP1:attr.1">open</a>
 
<b>by </b><i><a class="txt" href="complsp1.html#E1:109_1_1_1_1"><i><font color="Green" title="E85">A1</font></i></a>, <a class="ref" href="complsp1.html#T102" target="_self" title="COMPLSP1:th.102">Th102</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">z</font> =  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">A</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<a NAME="E4:109_1_1_1_1"/>
<font color="Maroon" title="c5">z</font> is <a href="complsp1.html#V1" title="COMPLSP1:attr.1">open</a>
 
<b>by </b><i><a class="txt" href="complsp1.html#E2:109_1_1_1_1"><i><font color="Green" title="E86">A2</font></i></a>, <a class="ref" href="complsp1.html#T78" target="_self" title="COMPLSP1:th.78">Th78</a></i>;<br/>
<b>hence </b><a NAME="E5:109_1_1_1_1"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">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 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="complsp1.html#K22" title="COMPLSP1:func.22">ComplexOpenSets</a> <font color="Maroon" title="c1">n</font></span>)</span> #)
 ;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="complsp1.html#K22" title="COMPLSP1:func.22">ComplexOpenSets</a> <font color="Maroon" title="c1">n</font></span>)</span> #);<br/>

<b>assume </b><a NAME="E5:109_1_1_1"/><i><font color="Green" title="E85">A3</font></i>: 
( <font color="Maroon" title="c4">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 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="complsp1.html#K22" title="COMPLSP1:func.22">ComplexOpenSets</a> <font color="Maroon" title="c1">n</font></span>)</span> #) &amp; <font color="Maroon" title="c5">B</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 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="complsp1.html#K22" title="COMPLSP1:func.22">ComplexOpenSets</a> <font color="Maroon" title="c1">n</font></span>)</span> #) )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c6">z1</font> = <font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c7">z2</font> = <font color="Maroon" title="c5">B</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<a NAME="E7:109_1_1_1"/><i><font color="Green" title="E86">A4</font></i>: 
( <font color="Maroon" title="c6">z1</font> is <a href="complsp1.html#V1" title="COMPLSP1:attr.1">open</a> &amp; <font color="Maroon" title="c7">z2</font> is <a href="complsp1.html#V1" title="COMPLSP1:attr.1">open</a> )
 
<b>by </b><i><a class="txt" href="complsp1.html#E5:109_1_1_1"><i><font color="Green" title="E85">A3</font></i></a>, <a class="ref" href="complsp1.html#T102" target="_self" title="COMPLSP1:th.102">Th102</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">z</font> = <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">B</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<a NAME="E9:109_1_1_1"/>
<font color="Maroon" title="c8">z</font> is <a href="complsp1.html#V1" title="COMPLSP1:attr.1">open</a>
 
<b>by </b><i><a class="txt" href="complsp1.html#E7:109_1_1_1"><i><font color="Green" title="E86">A4</font></i></a>, <a class="ref" href="complsp1.html#T79" target="_self" title="COMPLSP1:th.79">Th79</a></i>;<br/>
<b>hence </b><a NAME="E10:109_1_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">B</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 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="complsp1.html#K22" title="COMPLSP1:func.22">ComplexOpenSets</a> <font color="Maroon" title="c1">n</font></span>)</span> #)
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:109_1_1"/>
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="complsp1.html#K22" title="COMPLSP1:func.22">ComplexOpenSets</a> <font color="Maroon" title="c1">n</font></span>)</span> #) is  <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>
 ;<br/>


</div>
