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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>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>] means <font color="Maroon">a<sub>1</sub></font> <a href="waybel11.html#NR1" target="_self">has_the_property_(S)</a> ;<br/>
<b>consider </b><font color="Maroon">c<sub>2</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>, <font color="Maroon">c<sub>3</sub></font> being  <a href="pre_topc.html#V2">correct</a> <a href="waybel19.html#V1" target="_self">lower</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E2:54"/><i><font color="Green">E37</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>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</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> #) &amp;  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>3</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/>
<a NAME="E3:54"/><i><font color="Green">E38</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="waybel19.html#K1" target="_self">omega</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="waybel19.html#D3" target="_self">Def3</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</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>2</sub></font> <b>by </b><i><a class="txt" href="waybel19.html#E2:54"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</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>3</sub></font> <b>by </b><i><a class="txt" href="waybel19.html#E2:54"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>assume </b><a NAME="E4:54_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E5:54_1"/><b>then </b>
( ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>2</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel19.html#K1" target="_self">omega</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="waybel19.html#K1" target="_self">omega</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>3</sub></font> ) )
 
<b>by </b><i><a class="ref" href="waybel19.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="yellow_9.html#T51">YELLOW_9:51</a></i>;<br/>
<a NAME="E6:54_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a> or <font color="Maroon">c<sub>7</sub></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/>
<a NAME="E7:54_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="waybel11.html#NR1" target="_self">has_the_property_(S)</a>  or <font color="Maroon">c<sub>7</sub></font> <a href="waybel11.html#NR1" target="_self">has_the_property_(S)</a>  )
 
<b>by </b><i><a class="ref" href="waybel19.html#T35" target="_self">Th35</a>, <a class="ref" href="waybel11.html#T15">WAYBEL11:15</a></i>;<br/>
<b>hence </b><a NAME="E8:54_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>]
 <b>by </b><i><a class="txt" href="waybel19.html#E2:54"><i><font color="Green">E37</font></i></a>, <a class="ref" href="yellow12.html#T19">YELLOW12:19</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:54"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> st ( for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="setfam_1.html#K5">union</a> <font color="Olive">b<sub>1</sub></font>]
 
<b>by </b><i><a class="txt" href="waybel19.html#E41"><i><font color="Green">Lemma32</font></i></a></i>;<br/>
<a NAME="E5:54"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>2</sub></font>]
 
<b>by </b><i><a class="txt" href="waybel19.html#E42"><i><font color="Green">Lemma33</font></i></a></i>;<br/>
<a NAME="E6:54"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>]
 
;<br/>
<b>thus </b><a NAME="E7:54"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="waybel19.html#S1" target="_self">WAYBEL19:sch 1</a>(<a class="txt" href="waybel19.html#E3:54"><i><font color="Green">E38</font></i></a>, <a class="txt" href="waybel19.html#E4:54"><i><font color="Green">E39</font></i></a>, <a class="txt" href="waybel19.html#E5:54"><i><font color="Green">E40</font></i></a>, <a class="txt" href="waybel19.html#E6:54"><i><font color="Green">E41</font></i></a>);</i><br/>


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