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<div class="add">

<b>set </b><font color="Maroon">T</font> = <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Olive">S</font> is <a href="waybel_0.html#V13">upper</a> &amp; <font color="Olive">S</font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a> ) </span>}</span> ;<br/>
<a NAME="E1:19_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Olive">S</font> is <a href="waybel_0.html#V13">upper</a> &amp; <font color="Olive">S</font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:19_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Olive">S</font> is <a href="waybel_0.html#V13">upper</a> &amp; <font color="Olive">S</font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">S</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:19_1_1_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> &amp; <font color="Maroon">S</font> is <a href="waybel_0.html#V13">upper</a> &amp; <font color="Maroon">S</font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a> )
 ;<br/>
<b>thus </b><a NAME="E4:19_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">T</font> = <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Olive">S</font> is <a href="waybel_0.html#V13">upper</a> &amp; <font color="Olive">S</font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a> ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">L</font> ;<br/>
<b>set </b><font color="Maroon">SL</font> =  <a href="waybel_9.html#G1">TopRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font>,<font color="Maroon">T</font> #);<br/>
<a NAME="E3:19_1_1"/><i><font color="Green">E47</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <a href="waybel_9.html#G1">TopRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font>,<font color="Maroon">T</font> #),the <a href="orders_2.html#U1">InternalRel</a> of <a href="waybel_9.html#G1">TopRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font>,<font color="Maroon">T</font> #) #)
 
;<br/>
<b>then reconsider </b><font color="Maroon">SL</font> =  <a href="waybel_9.html#G1">TopRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font>,<font color="Maroon">T</font> #) as    <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon">SL</font>
;<br/>

<a NAME="E5:19_1_1"/>
for <font color="Olive">S</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">SL</font> holds <br/> ( <font color="Olive">S</font> is <a href="pre_topc.html#V3">open</a> iff ( <font color="Olive">S</font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a> &amp; <font color="Olive">S</font> is <a href="waybel_0.html#V13">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">S</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">SL</font>;<br/>

<b>thus </b><a NAME="E1:19_1_1_2"/>
( <font color="Maroon">S</font> is <a href="pre_topc.html#V3">open</a> implies ( <font color="Maroon">S</font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon">S</font> is <a href="waybel_0.html#V13">upper</a> ) )
 <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">S</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:19_1_1_2_1"/><b>then </b>
<font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">S1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:19_1_1_2_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> &amp; <font color="Maroon">S1</font> is <a href="waybel_0.html#V13">upper</a> &amp; <font color="Maroon">S1</font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a> )
 ;<br/>
<b>thus </b><a NAME="E5:19_1_1_2_1"/>
( <font color="Maroon">S</font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon">S</font> is <a href="waybel_0.html#V13">upper</a> )
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#T25">WAYBEL_0:25</a>, <a class="ref" href="yellow_9.html#T47">YELLOW_9:47</a></i>;<br/>


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

<b>thus </b><a NAME="E2:19_1_1_2"/>
( <font color="Maroon">S</font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon">S</font> is <a href="waybel_0.html#V13">upper</a> implies <font color="Maroon">S</font> is <a href="pre_topc.html#V3">open</a> )
 <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">S1</font> = <font color="Maroon">S</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/>
<b>assume </b><a NAME="E2:19_1_1_2_2"/>
( <font color="Maroon">S</font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon">S</font> is <a href="waybel_0.html#V13">upper</a> )
 ;<br/>

<a NAME="E3:19_1_1_2_2"/><b>then </b>
( <font color="Maroon">S1</font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon">S1</font> is <a href="waybel_0.html#V13">upper</a> )
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T25">WAYBEL_0:25</a>, <a class="ref" href="yellow_9.html#T47">YELLOW_9:47</a></i>;<br/>
<a NAME="E4:19_1_1_2_2"/><b>then </b>
<font color="Maroon">S</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font>
 
;<br/>
<b>hence </b><a NAME="E5:19_1_1_2_2"/>
<font color="Maroon">S</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/>


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


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


</div>
