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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:41"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V12">lower</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  <a href="waybel_0.html#V12">lower</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:41"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="waybel_0.html#V13">upper</a>
 
;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E5:41"/><i><font color="Green">E30</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></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 <font color="Maroon">c<sub>4</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>4</sub></font> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> ;<br/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="waybel28.html#E5:41"><i><font color="Green">E30</font></i></a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>4</sub></font>

;<br/>
<b>then </b><i><font color="Green">E31</font></i>: <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  = 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>6</sub></font>

<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K3">`</a> 

;<br/>
<b>thus </b><a NAME="E10:41"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a>  <a href="waybel28.html#K4" target="_self">xi</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>2</sub></font> is <a href="waybel11.html#V2">closed_under_directed_sups</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a>  <a href="waybel28.html#K4" target="_self">xi</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:41_3"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel28.html#E3:41"><i><font color="Green">E29</font></i></a>, <a class="txt" href="waybel28.html#E9:41"><i><font color="Green">E31</font></i></a>, <a class="ref" href="waybel28.html#T30" target="_self">Th30</a></i>;<br/>
<a NAME="E3:41_3"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T51">YELLOW_9:51</a></i>;<br/>
<a NAME="E4:41_3"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K3">`</a>  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="E5:41_3"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>
<a NAME="E6:41_3"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is <a href="waybel11.html#V2">closed_under_directed_sups</a>
 
<b>by </b><i><a class="ref" href="waybel11.html#T7">WAYBEL11:7</a></i>;<br/>
<b>hence </b><a NAME="E7:41_3"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel11.html#V2">closed_under_directed_sups</a>
 <b>by </b><i><a class="txt" href="waybel28.html#E5:41"><i><font color="Green">E30</font></i></a>, <a class="ref" href="yellow12.html#T20">YELLOW12:20</a></i>;<br/>


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

<b>assume </b><a NAME="E11:41"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel11.html#V2">closed_under_directed_sups</a>
 ;<br/>

<a NAME="E12:41"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="waybel11.html#V2">closed_under_directed_sups</a>
 
<b>by </b><i><a class="txt" href="waybel28.html#E5:41"><i><font color="Green">E30</font></i></a>, <a class="ref" href="yellow12.html#T20">YELLOW12:20</a></i>;<br/>
<a NAME="E13:41"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="waybel_0.html#V12">lower</a>
 
<b>by </b><i><a class="txt" href="waybel28.html#E1:41"><i><font color="Green">E28</font></i></a>, <a class="txt" href="waybel28.html#E5:41"><i><font color="Green">E30</font></i></a>, <a class="ref" href="waybel_0.html#T25">WAYBEL_0:25</a></i>;<br/>
<a NAME="E14:41"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="waybel28.html#E12:41"><i><font color="Green">E32</font></i></a>, <a class="ref" href="waybel11.html#T7">WAYBEL11:7</a></i>;<br/>
<a NAME="E15:41"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>
<a NAME="E16:41"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E17:41"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel28.html#E9:41"><i><font color="Green">E31</font></i></a>, <a class="ref" href="yellow_9.html#T51">YELLOW_9:51</a></i>;<br/>
<b>hence </b><a NAME="E18:41"/>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a>  <a href="waybel28.html#K4" target="_self">xi</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="waybel28.html#T29" target="_self">Th29</a></i>;<br/>


</div>
