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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel19.html#V2" target="_self">Lawson</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <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/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:60_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/><a NAME="E2:60_1"/><b>then </b>
<font color="Maroon">c<sub>2</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/><b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  as  <a href="waybel11.html#V3">property(S)</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="waybel19.html#T36" target="_self">Th36</a></i>;<br/><a NAME="E4:60_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>hence </b><a NAME="E5:60_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel11.html#V2">closed_under_directed_sups</a>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:60"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel11.html#V2">closed_under_directed_sups</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="waybel11.html#V2">directly_closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<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="E5:60"/><i><font color="Green">E41</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>4</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>4</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>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</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>3</sub></font> <a href="subset_1.html#K3">`</a>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> ;<br/>
<a NAME="E7:60"/>
( <font color="Maroon">c<sub>5</sub></font> is <a href="waybel_0.html#V13">upper</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a> )
 
<b>by </b><i><a class="txt" href="waybel19.html#E5:60"><i><font color="Green">E41</font></i></a>, <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="E8:60"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>1</sub></font> is    <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel19.html#E5:60"><i><font color="Green">E41</font></i></a>, <a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a>, <a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<a NAME="E9:60"/><b>then </b>
<font color="Maroon">c<sub>3</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="waybel19.html#T37" target="_self">Th37</a></i>;<br/>
<b>hence </b><a NAME="E10:60"/>
<font color="Maroon">c<sub>2</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/>


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