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<b>let </b><font color="Maroon" title="c1">N</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_2.html#V2" title="WAYBEL_2:attr.2">meet-continuous</a> <a href="waybel19.html#V2" title="WAYBEL19:attr.2">Lawson</a> <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a>;<br/>

<b>consider </b><font color="Maroon" title="c2">S</font> being  <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>  <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">N</font>;<br/>
<a NAME="E2:64"/><i><font color="Green" title="E34">A1</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">S</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">N</font> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:64_1"/><i><font color="Green" title="E35">A2</font></i>: 
<font color="Maroon" title="c1">N</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c3">M</font> = <font color="Maroon" title="c1">N</font> as  <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> ;<br/><a NAME="E3:64_1"/>
<font color="Maroon" title="c3">M</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>
 ;<br/><b>hence </b><a NAME="E4:64_1"/>
<font color="Maroon" title="c1">N</font> is <a href="waybel30.html#V3" title="WAYBEL30:attr.3">with_open_semilattices</a>
 <b>by </b><i><a class="ref" href="waybel30.html#T35" target="_self" title="WAYBEL30:th.35">Th35</a></i>;<br/><a NAME="E5:64_1"/>
<font color="Maroon" title="c2">S</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a>
 <b>by </b><i><a class="txt" href="waybel30.html#E2:64"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="waybel30.html#E1:64_1"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="waybel_8.html#T17" title="WAYBEL_8:th.17">WAYBEL_8:17</a></i>;<br/><a NAME="E6:64_1"/><b>then </b>
 ex <font color="Olive" title="b1">K</font> being   <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c2">S</font> st <font color="Olive" title="b1">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b2">x</font></span>)</span> where <font color="Olive" title="b2">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font> : <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c2">S</font></span>)</span> </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel14.html#T42" title="WAYBEL14:th.42">WAYBEL14:42</a></i>;<br/><a NAME="E7:64_1"/><b>then </b>
 <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c2">S</font></span>)</span> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a>
 <b>by </b><i><a class="ref" href="waybel14.html#T43" title="WAYBEL14:th.43">WAYBEL14:43</a></i>;<br/><b>hence </b><a NAME="E8:64_1"/>
 <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">N</font></span>)</span> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a>
 <b>by </b><i><a class="txt" href="waybel30.html#E2:64"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="yellow_9.html#T52" title="YELLOW_9:th.52">YELLOW_9:52</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><b>that </b><br/><a NAME="E4:64"/><i><font color="Green" title="E35">A3</font></i>: 
<font color="Maroon" title="c1">N</font> is <a href="waybel30.html#V3" title="WAYBEL30:attr.3">with_open_semilattices</a>
 <b>and </b><br/><a NAME="E5:64"/><i><font color="Green" title="E36">A4</font></i>: 
 <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">N</font></span>)</span> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c3">T</font> =  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">N</font></span>)</span> as  <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> <b>by </b><i><a class="txt" href="waybel30.html#E5:64"><i><font color="Green" title="E36">A4</font></i></a></i>;<br/>
<a NAME="E7:64"/>
<font color="Maroon" title="c3">T</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>
 
;<br/>
<a NAME="E8:64"/><b>then </b>
<font color="Maroon" title="c1">N</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>
 
<b>by </b><i><a class="txt" href="waybel30.html#E4:64"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="waybel30.html#T35" target="_self" title="WAYBEL30:th.35">Th35</a></i>;<br/>
<a NAME="E9:64"/><b>then </b>
<font color="Maroon" title="c2">S</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>
 
<b>by </b><i><a class="txt" href="waybel30.html#E2:64"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="yellow12.html#T15" title="YELLOW12:th.15">YELLOW12:15</a></i>;<br/>
<a NAME="E10:64"/><b>then </b>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>  ex <font color="Olive" title="b2">K</font> being   <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Olive" title="b1">x</font> st <br/>for <font color="Olive" title="b3">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b3">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">K</font> holds <br/>( <font color="Olive" title="b3">Y</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">Y</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> )
 
<b>by </b><i><a class="ref" href="waybel14.html#T35" title="WAYBEL14:th.35">WAYBEL14:35</a></i>;<br/>
<a NAME="E11:64"/><b>then </b><i><font color="Green" title="E37">A5</font></i>: 
for <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>  ex <font color="Olive" title="b2">VV</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b2">VV</font> &amp; ( for <font color="Olive" title="b3">W</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>  st <font color="Olive" title="b3">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">VV</font> holds <br/><font color="Olive" title="b3">W</font> is <a href="waybel_6.html#V6" title="WAYBEL_6:attr.6">co-prime</a> ) )
 
<b>by </b><i><a class="ref" href="waybel14.html#T39" title="WAYBEL14:th.39">WAYBEL14:39</a></i>;<br/>
<a NAME="E12:64"/>
 <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c2">S</font></span>)</span> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a>
 
<b>by </b><i><a class="txt" href="waybel30.html#E2:64"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="waybel30.html#E5:64"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="yellow_9.html#T52" title="YELLOW_9:th.52">YELLOW_9:52</a></i>;<br/>
<a NAME="E13:64"/><b>then </b>
 ex <font color="Olive" title="b1">K</font> being   <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c2">S</font> st <font color="Olive" title="b1">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b2">x</font></span>)</span> where <font color="Olive" title="b2">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font> : <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c2">S</font></span>)</span> </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel30.html#E11:64"><i><font color="Green" title="E37">A5</font></i></a>, <a class="ref" href="waybel14.html#T44" title="WAYBEL14:th.44">WAYBEL14:44</a></i>;<br/>
<a NAME="E14:64"/><b>then </b>
<font color="Maroon" title="c2">S</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a>
 
<b>by </b><i><a class="ref" href="waybel14.html#T45" title="WAYBEL14:th.45">WAYBEL14:45</a></i>;<br/>
<b>hence </b><a NAME="E15:64"/>
<font color="Maroon" title="c1">N</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a>
 <b>by </b><i><a class="txt" href="waybel30.html#E2:64"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="waybel_8.html#T17" title="WAYBEL_8:th.17">WAYBEL_8:17</a></i>;<br/>


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