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

<b>let </b><font color="Maroon">IT</font> be    <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E1:18_2_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">IT</font> is <a href="waybel11.html#V4">Scott</a>
 ;<br/>

<a NAME="E2:18_2_1"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">IT</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a></i>;<br/>
<a NAME="E3:18_2_1"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">IT</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">IT</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="E4:18_2_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IT</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">IT</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a><br/>

<b>thus </b><a NAME="E5:18_2_1"/>
for <font color="Olive">a</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">IT</font>  st <font color="Olive">a</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">IT</font> holds <br/> <a href="setfam_1.html#K5">union</a> <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">IT</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_1"><b>proof </b></a><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">IT</font>;<br/>

<b>assume </b><a NAME="E1:18_2_1_1"/><i><font color="Green">E47</font></i>: 
<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">IT</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">IT</font>;<br/><b>assume </b><a NAME="E1:18_2_1_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E2:18_2_1_1_1"/><b>then </b>
<font color="Maroon">X</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>hence </b><a NAME="E3:18_2_1_1_1"/>
<font color="Maroon">X</font> is <a href="waybel_0.html#V13">upper</a>
 <b>by </b><i>, <a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:18_2_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">a</font> is <a href="waybel_0.html#V13">upper</a>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T28">WAYBEL_0:28</a></i>;<br/>
<a NAME="E4:18_2_1_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">a</font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">IT</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D1">WAYBEL11:def 1</a><br/>

<b>assume </b><a NAME="E1:18_2_1_1_2"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">a</font>
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:18_2_1_1_2"/><i><font color="Green">E50</font></i>: 
(  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">IT</font> <b>by </b><i/>;<br/>
<a NAME="E5:18_2_1_1_2"/>
<font color="Maroon">A</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/>
<a NAME="E6:18_2_1_1_2"/><b>then </b>
<font color="Maroon">A</font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a>
 
<b>by </b><i>, <a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a></i>;<br/>
<a NAME="E7:18_2_1_1_2"/><b>then </b>
<font color="Maroon">D</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="waybel11.html#D1">WAYBEL11:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:18_2_1_1_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E10:18_2_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E11:18_2_1_1_2"/>
<font color="Maroon">D</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:18_2_1_1"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">a</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, , <a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:18_2_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">IT</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>


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

<b>thus </b><a NAME="E6:18_2_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">IT</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">IT</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">IT</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">IT</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_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">IT</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">IT</font>;<br/>

<b>assume </b><a NAME="E1:18_2_1_2"/><i><font color="Green">E54</font></i>: 
( <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">IT</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">IT</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font>, <font color="Maroon">b1</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">IT</font> ;<br/>
<a NAME="E3:18_2_1_2"/>
( <font color="Maroon">a1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">b1</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/>
<a NAME="E4:18_2_1_2"/><b>then </b><i><font color="Green">E72</font></i>: 
( <font color="Maroon">a1</font> is <a href="waybel_0.html#V13">upper</a> &amp; <font color="Maroon">a1</font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon">b1</font> is <a href="waybel_0.html#V13">upper</a> &amp; <font color="Maroon">b1</font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a> )
 
<b>by </b><i>, <a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a></i>;<br/>
<a NAME="E5:18_2_1_2"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">a</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">b</font> is <a href="waybel_0.html#V13">upper</a>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T29">WAYBEL_0:29</a></i>;<br/>
<a NAME="E6:18_2_1_2"/>
<font color="Maroon">a</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">b</font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">IT</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D1">WAYBEL11:def 1</a><br/>

<b>assume </b><a NAME="E1:18_2_1_2_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:18_2_1_2_1"/><b>then </b>
(  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a1</font> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:18_2_1_2_1"/><b>then </b><i><font color="Green">E75</font></i>: 
( <font color="Maroon">D</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">D</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">b1</font> )
 
<b>by </b><i>, <a class="ref" href="waybel11.html#D1">WAYBEL11:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">d1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:18_2_1_2_1"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">d1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">d1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a1</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon">d2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:18_2_1_2_1"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">d2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">d2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">d1</font> = <font color="Maroon">d1</font>, <font color="Maroon">d2</font> = <font color="Maroon">d2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">IT</font> <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">IT</font><b> such that </b><br/><a NAME="E10:18_2_1_2_1"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">d1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">d2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a></i>;<br/>
<a NAME="E11:18_2_1_2_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> )
 
<b>by </b><i>, , , , <a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/>
<a NAME="E12:18_2_1_2_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E13:18_2_1_2_1"/>
<font color="Maroon">D</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">a</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:18_2_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">b</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, , <a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:18_2_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">IT</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>


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


</div>
