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<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_2.html#V2">meet-continuous</a> <a href="waybel19.html#V2">Lawson</a> <a href="waybel_9.html#NM1">TopLattice</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>;<br/>
<a NAME="E2:57"/><i><font color="Green">E33</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>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<a NAME="E3:57"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T52">YELLOW_9:52</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:57_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_3.html#V3">continuous</a>
 ;<br/><a NAME="E2:57_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel_3.html#V3">continuous</a>
 <b>by </b><i><a class="txt" href="waybel30.html#E2:57"><i><font color="Green">E33</font></i></a>, <a class="ref" href="yellow12.html#T15">YELLOW12:15</a></i>;<br/><a NAME="E3:57_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="yellow_8.html#M1">Basis</a> of <font color="Olive">b<sub>1</sub></font> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> is   <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E2:57_1_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V2">filtered</a> )
 <b>by </b><i><a class="txt" href="waybel30.html#E2:57_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="waybel14.html#T35">WAYBEL14:35</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>2</sub></font>;<br/>

<b>assume </b><a NAME="E3:57_1_1"/><i><font color="Green">E37</font></i>: 
<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="E4:57_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="waybel30.html#E2:57_1_1"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:57_1_1"/>
<font color="Maroon">c<sub>5</sub></font> is   <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel30.html#E2:57_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="waybel30.html#E3:57_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a>, <a class="ref" href="yellow_8.html#T21">YELLOW_8:21</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:57_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel30.html#V3" target="_self">with_open_semilattices</a>
 <b>by </b><i><a class="ref" href="waybel30.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E5:57_1"/>
 <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="waybel_3.html#V3">continuous</a>
 <b>by </b><i><a class="txt" href="waybel30.html#E2:57_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="waybel14.html#T36">WAYBEL14:36</a></i>;<br/><b>hence </b><a NAME="E6:57_1"/>
 <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="waybel_3.html#V3">continuous</a>
 <b>by </b><i><a class="txt" href="waybel30.html#E2:57"><i><font color="Green">E33</font></i></a>, <a class="ref" href="yellow_9.html#T52">YELLOW_9:52</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E5:57"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="waybel30.html#V3" target="_self">with_open_semilattices</a> &amp;  <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="waybel_3.html#V3">continuous</a> )
 ;<br/>

<a NAME="E6:57"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="yellow_8.html#M1">Basis</a> of <font color="Olive">b<sub>1</sub></font> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>3</sub></font> is <a href="waybel_0.html#V2">filtered</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel30.html#E2:57"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E3:57_2"/><i><font color="Green">E36</font></i>: 
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> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> holds <br/> <a href="yellow_0.html#K5">subrelstr</a> <font color="Olive">b<sub>1</sub></font> is <a href="yellow_0.html#V5">meet-inheriting</a>
 <b>by </b><i><a class="txt" href="waybel30.html#E5:57"><i><font color="Green">E35</font></i></a>, <a class="ref" href="waybel30.html#D4" target="_self">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span>  as    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="waybel30.html#T16" target="_self">Th16</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E5:57_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:57_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E8:57_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>
<a NAME="E9:57_2"/>
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>8</sub></font> is <a href="yellow_0.html#V5">meet-inheriting</a>
 
<b>by </b><i><a class="txt" href="waybel30.html#E3:57_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="waybel30.html#E8:57_2"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E10:57_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> is <a href="waybel_0.html#V2">filtered</a>
 
<b>by </b><i><a class="ref" href="yellow12.html#T26">YELLOW12:26</a></i>;<br/>
<a NAME="E11:57_2"/><b>then </b>
 <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">c<sub>8</sub></font> is <a href="waybel_0.html#V2">filtered</a>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T35">WAYBEL_0:35</a></i>;<br/>
<b>hence </b><a NAME="E12:57_2"/>
( <font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>7</sub></font> is <a href="waybel_0.html#V2">filtered</a> )
 <b>by </b><i><a class="txt" href="waybel30.html#E2:57"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel30.html#E5:57_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="waybel30.html#E7:57_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="waybel_0.html#T4">WAYBEL_0:4</a>, <a class="ref" href="yellow_8.html#T21">YELLOW_8:21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:57"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> ,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="waybel30.html#E3:57"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel30.html#E5:57"><i><font color="Green">E35</font></i></a>, <a class="ref" href="waybel14.html#T37">WAYBEL14:37</a></i>;<br/>
<a NAME="E8:57"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel_3.html#V3">continuous</a>
 
<b>by </b><i><a class="ref" href="waybel14.html#T38">WAYBEL14:38</a></i>;<br/>
<b>hence </b><a NAME="E9:57"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_3.html#V3">continuous</a>
 <b>by </b><i><a class="txt" href="waybel30.html#E2:57"><i><font color="Green">E33</font></i></a>, <a class="ref" href="yellow12.html#T15">YELLOW12:15</a></i>;<br/>


</div>
