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

<b>let </b><font color="Maroon" title="c2">IT</font> be    <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">L</font>;<br/>

<b>assume </b><a NAME="E1:18_2_1"/><i><font color="Green" title="E10">A1</font></i>: 
<font color="Maroon" title="c2">IT</font> is <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>
 ;<br/>

<a NAME="E2:18_2_1"/><b>then </b>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">IT</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="waybel11.html#D4" title="WAYBEL11:def.4">WAYBEL11:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:18_2_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">IT</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="c2">IT</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>; <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="E4:18_2_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 <font color="Maroon" title="c2">IT</font>  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 <font color="Maroon" title="c2">IT</font> 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 <font color="Maroon" title="c2">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" title="c3">a</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">IT</font>;<br/>

<b>assume </b><a NAME="E1:18_2_1_1"/><i><font color="Green" title="E11">A2</font></i>: 
<font color="Maroon" title="c3">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 <font color="Maroon" title="c2">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" title="c4">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">IT</font>;<br/><b>assume </b><a NAME="E1:18_2_1_1_1"/>
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">a</font>
 ;<br/><a NAME="E2:18_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">X</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="waybel29.html#E1:18_2_1_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/><b>hence </b><a NAME="E3:18_2_1_1_1"/>
<font color="Maroon" title="c4">X</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>
 <b>by </b><i><a class="txt" href="waybel29.html#E1:18_2_1"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="waybel11.html#D4" title="WAYBEL11:def.4">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" title="E12">A3</font></i>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">a</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T28" title="WAYBEL_0:th.28">WAYBEL_0:28</a></i>;<br/>
<a NAME="E4:18_2_1_1"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">a</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">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" title="c4">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">IT</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D1" title="WAYBEL11:def.1">WAYBEL11:def 1</a><br/>

<b>assume </b><a NAME="E1:18_2_1_1_2"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">a</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:18_2_1_1_2"/><i><font color="Green" title="E13">A4</font></i>: 
(  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">a</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">A</font> = <font color="Maroon" title="c5">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">IT</font> <b>by </b><i><a class="txt" href="waybel29.html#E3:18_2_1_1_2"><i><font color="Green" title="E13">A4</font></i></a></i>;<br/>
<a NAME="E5:18_2_1_1_2"/>
<font color="Maroon" title="c6">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="waybel29.html#E1:18_2_1_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="waybel29.html#E3:18_2_1_1_2"><i><font color="Green" title="E13">A4</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="E6:18_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">A</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a>
 
<b>by </b><i><a class="txt" href="waybel29.html#E1:18_2_1"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="waybel11.html#D4" title="WAYBEL11:def.4">WAYBEL11:def 4</a></i>;<br/>
<a NAME="E7:18_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">D</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">A</font>
 
<b>by </b><i><a class="txt" href="waybel29.html#E3:18_2_1_1_2"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="waybel11.html#D1" title="WAYBEL11:def.1">WAYBEL11:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:18_2_1_1_2"/><i><font color="Green" title="E14">A5</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E10:18_2_1_1_2"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">a</font> )
 
<b>by </b><i><a class="txt" href="waybel29.html#E3:18_2_1_1_2"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="waybel29.html#E9:18_2_1_1_2"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E11:18_2_1_1_2"/>
<font color="Maroon" title="c4">D</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">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" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">a</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="waybel29.html#E1:18_2_1"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="waybel29.html#E3:18_2_1_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="waybel11.html#D4" title="WAYBEL11:def.4">WAYBEL11:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:18_2_1_1"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">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="c2">IT</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>


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

<b>thus </b><a NAME="E5:18_2_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">IT</font>  st <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 <font color="Maroon" title="c2">IT</font> &amp; <font color="Olive" title="b2">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 <font color="Maroon" title="c2">IT</font> holds <br/><font color="Olive" title="b1">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">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 <font color="Maroon" title="c2">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" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">IT</font>;<br/>

<b>assume </b><a NAME="E1:18_2_1_2"/><i><font color="Green" title="E11">A6</font></i>: 
( <font color="Maroon" title="c3">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="c2">IT</font> &amp; <font color="Maroon" title="c4">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 <font color="Maroon" title="c2">IT</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">a1</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c6">b1</font> = <font color="Maroon" title="c4">b</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">IT</font> ;<br/>
<a NAME="E3:18_2_1_2"/>
( <font color="Maroon" title="c5">a1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c6">b1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="txt" href="waybel29.html#E1:18_2_1_2"><i><font color="Green" title="E11">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="E4:18_2_1_2"/><b>then </b><i><font color="Green" title="E12">A7</font></i>: 
( <font color="Maroon" title="c5">a1</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> &amp; <font color="Maroon" title="c5">a1</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon" title="c6">b1</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> &amp; <font color="Maroon" title="c6">b1</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> )
 
<b>by </b><i><a class="txt" href="waybel29.html#E1:18_2_1"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="waybel11.html#D4" title="WAYBEL11:def.4">WAYBEL11:def 4</a></i>;<br/>
<a NAME="E5:18_2_1_2"/><b>then </b><i><font color="Green" title="E13">A8</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">b</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T29" title="WAYBEL_0:th.29">WAYBEL_0:29</a></i>;<br/>
<a NAME="E6:18_2_1_2"/>
<font color="Maroon" title="c3">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">b</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">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" title="c7">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">IT</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D1" title="WAYBEL11:def.1">WAYBEL11:def 1</a><br/>

<b>assume </b><a NAME="E1:18_2_1_2_1"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c7">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">b</font>
 ;<br/>

<a NAME="E2:18_2_1_2_1"/><b>then </b>
(  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c7">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">a1</font> &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c7">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">b1</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:18_2_1_2_1"/><b>then </b><i><font color="Green" title="E14">A9</font></i>: 
( <font color="Maroon" title="c7">D</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c5">a1</font> &amp; <font color="Maroon" title="c7">D</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">b1</font> )
 
<b>by </b><i><a class="txt" href="waybel29.html#E4:18_2_1_2"><i><font color="Green" title="E12">A7</font></i></a>, <a class="ref" href="waybel11.html#D1" title="WAYBEL11:def.1">WAYBEL11:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">d1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:18_2_1_2_1"/><i><font color="Green" title="E15">A10</font></i>: 
( <font color="Maroon" title="c8">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font> &amp; <font color="Maroon" title="c8">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">a1</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">d2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:18_2_1_2_1"/><i><font color="Green" title="E16">A11</font></i>: 
( <font color="Maroon" title="c9">d2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font> &amp; <font color="Maroon" title="c9">d2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">b1</font> )
 <b>by </b><i><a class="txt" href="waybel29.html#E3:18_2_1_2_1"><i><font color="Green" title="E14">A9</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">d1</font> = <font color="Maroon" title="c8">d1</font>, <font color="Maroon" title="c11">d2</font> = <font color="Maroon" title="c9">d2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">IT</font> <b>by </b><i><a class="txt" href="waybel29.html#E5:18_2_1_2_1"><i><font color="Green" title="E15">A10</font></i></a>, <a class="txt" href="waybel29.html#E7:18_2_1_2_1"><i><font color="Green" title="E16">A11</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">IT</font><b> such that </b><br/><a NAME="E10:18_2_1_2_1"/><i><font color="Green" title="E17">A12</font></i>: 
( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font> &amp; <font color="Maroon" title="c10">d1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">z</font> &amp; <font color="Maroon" title="c11">d2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">z</font> )
 <b>by </b><i><a class="txt" href="waybel29.html#E5:18_2_1_2_1"><i><font color="Green" title="E15">A10</font></i></a>, <a class="txt" href="waybel29.html#E7:18_2_1_2_1"><i><font color="Green" title="E16">A11</font></i></a>, <a class="ref" href="waybel_0.html#D1" title="WAYBEL_0:def.1">WAYBEL_0:def 1</a></i>;<br/>
<a NAME="E11:18_2_1_2_1"/>
( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">a1</font> &amp; <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">b1</font> )
 
<b>by </b><i><a class="txt" href="waybel29.html#E4:18_2_1_2"><i><font color="Green" title="E12">A7</font></i></a>, <a class="txt" href="waybel29.html#E5:18_2_1_2_1"><i><font color="Green" title="E15">A10</font></i></a>, <a class="txt" href="waybel29.html#E7:18_2_1_2_1"><i><font color="Green" title="E16">A11</font></i></a>, <a class="txt" href="waybel29.html#E10:18_2_1_2_1"><i><font color="Green" title="E17">A12</font></i></a>, <a class="ref" href="waybel_0.html#D20" title="WAYBEL_0:def.20">WAYBEL_0:def 20</a></i>;<br/>
<a NAME="E12:18_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">b</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E13:18_2_1_2_1"/>
<font color="Maroon" title="c7">D</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="txt" href="waybel29.html#E10:18_2_1_2_1"><i><font color="Green" title="E17">A12</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:18_2_1_2"/><b>then </b>
<font color="Maroon" title="c3">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">b</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="waybel29.html#E1:18_2_1"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="waybel29.html#E5:18_2_1_2"><i><font color="Green" title="E13">A8</font></i></a>, <a class="ref" href="waybel11.html#D4" title="WAYBEL11:def.4">WAYBEL11:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:18_2_1_2"/>
<font color="Maroon" title="c3">a</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">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 <font color="Maroon" title="c2">IT</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>


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


</div>
