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

<b>set </b><font color="Maroon" title="c2">T</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">S</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> &amp; <font color="Olive" title="b1">S</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> ) </span>}</span> ;<br/>
<a NAME="E1:19_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">S</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> &amp; <font color="Olive" title="b1">S</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</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">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">S</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> &amp; <font color="Olive" title="b1">S</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> ) </span>}</span>  or <font color="Maroon" title="c3">x</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">L</font> )</span><br/>

<b>assume </b><a NAME="E1:19_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">S</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> &amp; <font color="Olive" title="b1">S</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</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="c3">x</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">L</font></span><br/>

<b>then consider </b><font color="Maroon" title="c4">S</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:19_1_1_1"/><i><font color="Green" title="E10">A1</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">S</font> &amp; <font color="Maroon" title="c4">S</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> &amp; <font color="Maroon" title="c4">S</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> )
 ;<br/>
<b>thus </b><a NAME="E4:19_1_1_1"/>
<font color="Maroon" title="c3">x</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">L</font>
 <b>by </b><i><a class="txt" href="#E3:19_1_1_1"><i><font color="Green" title="E10">A1</font></i></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>then reconsider </b><font color="Maroon" title="c3">T</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">S</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> &amp; <font color="Olive" title="b1">S</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> ) </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>set </b><font color="Maroon" title="c4">SL</font> =  <a href="waybel_9.html#G1" title="WAYBEL_9:aggr.1">TopRelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c3">T</font> #);<br/>
<a NAME="E3:19_1_1"/><i><font color="Green" title="E10">A2</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="waybel_9.html#G1" title="WAYBEL_9:aggr.1">TopRelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c3">T</font> #),the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <a href="waybel_9.html#G1" title="WAYBEL_9:aggr.1">TopRelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c3">T</font> #) #)
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">SL</font> =  <a href="waybel_9.html#G1" title="WAYBEL_9:aggr.1">TopRelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c3">T</font> #) as    <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">SL</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">SL</font> is <a href="waybel_9.html#V1" title="WAYBEL_9:attr.1">strict</a> &amp; <font color="Maroon" title="c5">SL</font> is <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a> )</span><br/>

<a NAME="E5:19_1_1"/>
for <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">SL</font> holds <br/> ( <font color="Olive" title="b1">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> iff ( <font color="Olive" title="b1">S</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> &amp; <font color="Olive" title="b1">S</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">S</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">SL</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> iff ( <font color="Maroon" title="c6">S</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon" title="c6">S</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> ) )</span><br/>

<b>thus </b><a NAME="E1:19_1_1_2"/>
( <font color="Maroon" title="c6">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> implies ( <font color="Maroon" title="c6">S</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon" title="c6">S</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">S</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon" title="c6">S</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> implies <font color="Maroon" title="c6">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1_1_2_1"/>
<font color="Maroon" title="c6">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">S</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon" title="c6">S</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> )</span><br/>

<a NAME="E2:19_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c6">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</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/>
<b>then consider </b><font color="Maroon" title="c7">S1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E4:19_1_1_2_1"/><i><font color="Green" title="E11">A3</font></i>: 
( <font color="Maroon" title="c7">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">S</font> &amp; <font color="Maroon" title="c7">S1</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> &amp; <font color="Maroon" title="c7">S1</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> )
 ;<br/>
<b>thus </b><a NAME="E5:19_1_1_2_1"/>
( <font color="Maroon" title="c6">S</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon" title="c6">S</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> )
 <b>by </b><i><a class="txt" href="#E3:19_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="#E4:19_1_1_2_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="waybel_0.html#T25" title="WAYBEL_0:th.25">WAYBEL_0:25</a>, <a class="ref" href="yellow_9.html#T47" title="YELLOW_9:th.47">YELLOW_9:47</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>thus </b><a NAME="E2:19_1_1_2"/>
( <font color="Maroon" title="c6">S</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon" title="c6">S</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> implies <font color="Maroon" title="c6">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_2_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c7">S1</font> = <font color="Maroon" title="c6">S</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>assume </b><a NAME="E2:19_1_1_2_2"/>
( <font color="Maroon" title="c6">S</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon" title="c6">S</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a></span><br/>

<a NAME="E3:19_1_1_2_2"/><b>then </b>
( <font color="Maroon" title="c7">S1</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon" title="c7">S1</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> )
 
<b>by </b><i><a class="txt" href="#E3:19_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="waybel_0.html#T25" title="WAYBEL_0:th.25">WAYBEL_0:25</a>, <a class="ref" href="yellow_9.html#T47" title="YELLOW_9:th.47">YELLOW_9:47</a></i>;<br/>
<a NAME="E4:19_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c6">S</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="c5">SL</font>
 
;<br/>
<b>hence </b><a NAME="E5:19_1_1_2_2"/>
<font color="Maroon" title="c6">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><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>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:19_1_1"/>
( <font color="Maroon" title="c5">SL</font> is <a href="waybel_9.html#V1" title="WAYBEL_9:attr.1">strict</a> &amp; <font color="Maroon" title="c5">SL</font> is <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a> )
 <b>by </b><i><a class="ref" href="waybel11.html#D4" title="WAYBEL11:def.4">WAYBEL11:def 4</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>
